Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Normalize a direction vector.
- Distinguish scalar and vector projections.
- Check that the remaining component is perpendicular.
Before you start
Vectors, coordinate magnitude, and dot products.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A displacement is v = (6, 2) metres. A reference direction is u = (3, 4). Find v's signed component along u and its vector projection.
Why this math matters
A displacement can be resolved along axes that differ from the map's horizontal and vertical axes. Projection answers how much motion points along one chosen direction.
Set up the model
A useful answer starts with clear assumptions:
- Coordinates use perpendicular equal-scale axes.
- u specifies direction and is nonzero.
- The positive reference direction follows u, not −u.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find the part of a displacement along a chosen direction
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A displacement is v = (6, 2) metres. A reference direction is u = (3, 4). Find v's signed component along u and its vector projection.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Find a unit direction
|u| = 5; û = (3/5, 4/5)
Dividing by magnitude preserves direction while making the vector length one.
Compute both projections
v · û = 26/5 = 5.2 m; projᵤv = (26/25)(3, 4) = (3.12, 4.16) m
The scalar projection is signed length. Multiplying that length by the unit direction produces a vector.
Check the remainder
v − projᵤv = (2.88, −2.16); (2.88, −2.16) · (3, 4) = 0
A zero dot product confirms that the leftover vector is perpendicular to the chosen direction. Adding both components recovers v.
The result
The signed component is 5.2 m and the projection vector is (3.12, 4.16) m.
The projection's vertical component exceeds v's vertical component because the perpendicular remainder points partly downward. Components combine to the original displacement.
Common mistakes to catch
- Dividing by |u| gives a scalar projection; multiplying u requires division by |u|².
- A negative scalar component indicates opposite direction, not a negative magnitude.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Project (−3, 4) onto the positive x-direction (1, 0).
Show a hint
Keep the horizontal component.
Reveal answer and explanation
Scalar −3; vector (−3, 0)
The signed component is negative because it points opposite the positive x-axis.
Practice 2
Are (2, 1) and (1, −2) perpendicular?
Show a hint
Take their dot product.
Reveal answer and explanation
Yes
2(1) + 1(−2) = 0, and both vectors are nonzero.
Take the idea with you
Specify a direction and sign convention before reporting a component of motion.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Build a triangle with three right angles on a sphere
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.