A suitcase pulled diagonally can move along the floor while also becoming easier to lift. One pull produces two useful effects. Components let us describe those effects without pretending that two extra physical forces have appeared.
Components are a way to describe one vector using chosen axes. They are not extra forces or separate journeys that secretly happened. We will build them from right-triangle geometry, handle signs and quadrants, add vectors, and finally choose axes that simplify a physics problem. You need the sine and cosine ratios of a right triangle; the first section explains how we use them.
What you will learn to do
- Distinguish a vector, its magnitude and its signed components.
- Resolve a vector relative to the angle and axes actually specified.
- Add or subtract vectors component by component and reconstruct a direction.
- Use magnitude and limiting-angle checks to detect component errors.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Can two displacements of equal length add to zero? What information is missing from their lengths alone?
Compare your reasoning
Yes, if their directions are opposite. A vector needs direction as well as magnitude. Components preserve that directional information while allowing ordinary signed arithmetic.
Revisit the explanation →01 / Vectors and their components
What information does a vector contain?
A scalar is specified by a value with its units: mass, temperature or elapsed time. A vector also has direction: displacement, velocity, acceleration and force are examples. The magnitude of a vector is nonnegative. Its component along a chosen axis can be negative because the sign records direction.
Choose +x east and +y north. A displacement straight west has a negative x component and zero y component. That does not mean the distance travelled is negative. It means the change in position is opposite the positive x direction.
We draw a vector as an arrow whose orientation gives direction and whose length represents magnitude on a chosen scale. For adding displacement vectors, we can move an arrow parallel to itself without changing it. In mechanics, relocating the actual point where a force acts can change torque; the drawing rule for vector addition is not permission to move a physical force to any point on a rigid body.
The unit vectors point along the positive axes and have magnitude one. Ax and Ay carry the physical units and the signs.
STOP & THINK
A velocity has x component −4 m/s. Which statement must be true?
- Its speed is negative
- Its x motion is opposite +x
- It is necessarily slowing down
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components
Focus: What information does a vector contain?
Learning goals:
- Distinguish a vector, its magnitude and its signed components.
- Resolve a vector relative to the angle and axes actually specified.
- Add or subtract vectors component by component and reconstruct a direction.
- Use magnitude and limiting-angle checks to detect component errors.
Equations in this section (LaTeX):
\mathbf A=A_x\hat{\mathbf x}+A_y\hat{\mathbf y}
Meaning: The unit vectors point along the positive axes and have magnitude one. Ax and Ay carry the physical units and the signs.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Vectors and their components
Build the component formulas from a triangle
Place the tail of a vector at the origin and draw perpendicular projections from its tip to the axes. The horizontal projection is Ax and the vertical projection is Ay. Together they identify exactly the same endpoint as the original arrow.
If θ is measured counterclockwise from +x, the horizontal component is adjacent to that angle and the vertical component is opposite it. The triangle definitions give cos θ = adjacent/hypotenuse and sin θ = opposite/hypotenuse. Multiplying by the vector magnitude A gives the component formulas.
These formulas continue to work beyond the first quadrant because sine and cosine acquire signs. The vector magnitude A remains nonnegative while its projections move to the negative sides of the axes. Do not replace a negative component by its absolute value before adding vectors.
Here θ is explicitly measured counterclockwise from the positive x axis. An angle measured from a different axis requires different geometry.
INTERACTIVE LAB
Rotate one force, watch both projections
Before moving the angle, predict the signs in each quadrant. Find an angle where one component is zero without making the force itself zero.
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At θ = 0°, the vector lies entirely along +x: Ax = A and Ay = 0. At 90°, the roles exchange. At 180°, Ax = −A. These limiting cases are quick tests of whether you have assigned sine and cosine correctly.
An individual component cannot have a magnitude greater than A. If a 10 N force appears to have a 14 N horizontal component, the geometry or arithmetic is wrong. On perpendicular axes, the squared components add to the squared magnitude.
This is the Pythagorean relation, or equivalently sin² θ + cos² θ = 1. It is a check on both components together.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Focus: Build the component formulas from a triangle Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Equations in this section (LaTeX): A_x=A\cos\theta,\qquad A_y=A\sin\theta Meaning: Here θ is explicitly measured counterclockwise from the positive x axis. An angle measured from a different axis requires different geometry. A_x^2+A_y^2=A^2 Meaning: This is the Pythagorean relation, or equivalently sin² θ + cos² θ = 1. It is a check on both components together. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Vectors and their components
Get the sign before trusting the calculator
WORKED EXAMPLE
Resolve a force in the second quadrant
A 12 N force makes an angle of 150° counterclockwise from +x. Find its Cartesian components.
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The instruction “30° west of north” describes a different reference axis. Starting from +y, turn toward −x. The northward component is now adjacent to 30°, so it uses cosine; the westward component uses sine and has a negative x sign. Memorizing “x always uses cosine” without checking the angle definition causes avoidable errors.
STOP & THINK
A 10 N force points 30° east of north. What is its eastward component?
- 10 cos 30° N
- 10 sin 30° N
- −10 sin 30° N
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Focus: Get the sign before trusting the calculator Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Vectors and their components
Add directions, not just magnitudes
Two successive displacements add head to tail: the second begins where the first ends, and the resultant goes from the original start to the final end. Components perform the same construction numerically. Horizontal changes add to horizontal changes; vertical changes add to vertical changes.
Adding magnitudes is valid only for vectors pointing in the same direction. Opposite vectors subtract in magnitude, and perpendicular vectors require a right triangle. This is why total path length and displacement magnitude can differ.
WORKED EXAMPLE
Separate the journey from the displacement
A student walks 6 m east, then 8 m north. Find the distance travelled, the displacement magnitude and its direction.
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WORKED EXAMPLE
A resultant can be smaller than the sum of magnitudes
Two forces have components A = (4, 3) N and B = (−1, 1) N. Find their resultant.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Focus: Add directions, not just magnitudes Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Vectors and their components
A change in a vector is also a vector
Subtracting B means adding −B, the vector with the same magnitude as B but opposite direction. In components this is ordinary signed subtraction. Write brackets when subtracting a negative component to avoid accidentally reversing only part of a vector.
This operation is especially important for acceleration. The change in velocity is final velocity minus initial velocity, not final speed minus initial speed. A particle can keep the same speed while changing direction and therefore have nonzero acceleration.
WORKED EXAMPLE
Same speed, nonzero velocity change
A particle initially moves east at 5 m/s and later moves north at 5 m/s. The change takes 2 s. Find its average acceleration.
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STOP & THINK
An x velocity changes from −3 m/s to +5 m/s in 4 s. What is the average x acceleration?
Answer in m/s².
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Focus: A change in a vector is also a vector Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Vectors and their components
Recover a direction without losing its quadrant
The magnitude formula uses squares, so it cannot tell you which quadrant the original vector occupied. The component signs must be retained separately. Likewise, a calculator’s ordinary inverse tangent may return only one of several angles that have the same tangent.
For example, a vector with Ax = −3 and Ay = +4 is in the second quadrant. The ratio Ay/Ax is negative, and a basic inverse tangent may return a negative first/fourth-quadrant angle. The correct counterclockwise direction from +x is 180° − tan⁻¹(4/3), about 126.9°.
Use a sketch and the signs, or a two-argument atan2 function that explicitly accepts both components in the documented order. If Ax is zero, do not divide by zero to use a tangent formula: the sign of Ay immediately distinguishes +y from −y.
| Ax | Ay | Direction region |
|---|---|---|
| + | + | First quadrant |
| − | + | Second quadrant |
| − | − | Third quadrant |
| + | − | Fourth quadrant |
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components Focus: Recover a direction without losing its quadrant Learning goals: - Distinguish a vector, its magnitude and its signed components. - Resolve a vector relative to the angle and axes actually specified. - Add or subtract vectors component by component and reconstruct a direction. - Use magnitude and limiting-angle checks to detect component errors. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Vectors and their components
Choose axes to simplify the situation
The physical vector does not change when you rotate the coordinate axes, but its component values do. A force along a ramp may have awkward horizontal and vertical components yet only one nonzero component on an axis parallel to the ramp. Sensible axes simplify equations; they do not alter the physics.
For a ramp rising at angle θ above horizontal, choose one positive axis up the ramp and the other away from its surface. Gravity still points vertically down. Its parallel component is −mg sin θ, down the ramp, and its normal component is −mg cos θ, into the surface.
Test these assignments at θ = 0: a horizontal surface has no downslope gravity component, and the full weight acts perpendicular to it. At θ approaching 90°, nearly all the weight lies along the ramp. These limits are more convincing than remembering which formula appeared beside a particular diagram.
The signs refer to axes up the ramp and away from it. The original gravitational force remains one vertical vector.
Before using components in a force or motion equation, write the axis directions, identify the angle’s reference, and predict each sign. After calculation, reconstruct the magnitude or check a simple limiting angle. This short routine catches a large fraction of errors before they spread into the rest of a solution.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Vectors and their components
Focus: Choose axes to simplify the situation
Learning goals:
- Distinguish a vector, its magnitude and its signed components.
- Resolve a vector relative to the angle and axes actually specified.
- Add or subtract vectors component by component and reconstruct a direction.
- Use magnitude and limiting-angle checks to detect component errors.
Equations in this section (LaTeX):
W_{\parallel}=-mg\sin\theta,\qquad W_{\perp}=-mg\cos\theta
Meaning: The signs refer to axes up the ramp and away from it. The original gravitational force remains one vertical vector.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.BRING THE IDEAS TOGETHER
What should stay with you
- A vector has a nonnegative magnitude and a direction; components are signed projections on chosen axes.
- Cosine belongs to the component adjacent to the stated angle, not automatically to every quantity named x.
- Add and subtract signed components before finding magnitudes.
- Equal speeds do not imply equal velocities; direction changes contribute to acceleration.
- Changing axes changes components but preserves the physical vector and its magnitude.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A displacement has components (−9, 12) m. What is its magnitude?
Answer in m.
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SELF CHECK 2
Which quadrant contains a vector with components (−9, 12)?
- First
- Second
- Third
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SELF CHECK 3
For A = (2, −3) m and B = (−4, 1) m, what is the x component of A − B?
Answer in m.
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SELF CHECK 4
A vector points entirely along +y. What is its x component?
- Equal to its magnitude
- Zero
- Negative its magnitude
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
A displacement with two componentsBeginner · Free sample · Hints and guided reasoningCross the river, then measure the drift.Intermediate · Hints and guided reasoningA pull partly lifts the loadIntermediate · Hints and guided reasoningOnly part of the pull is horizontalIntermediate · Hints and guided reasoningA sideways component cancelsIntermediate · Hints and guided reasoningCrossing without being swept downstreamIntermediate · Hints and guided reasoningRain seen from a moving trolleyIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
A force stays fixed while you rotate your coordinate axes. What changes, and what must stay the same?
Compare your reasoning
Its component numbers generally change. Its physical direction and magnitude stay the same. Reconstructing the vector from the new components must give the original force.
Revisit the explanation →A few days later: change the situation
You walk 3 m east, then 4 m north. Explain the difference between distance walked and displacement magnitude.
Compare your reasoning
The distance is 7 m along the route. The displacement joins start to finish and has magnitude 5 m. Its components are 3 m east and 4 m north.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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