Vectors in space: magnitude, direction, and adding displacements

Where we are on the map

We move from the address of a point to a directed change between points.

Begin with a familiar image

The instruction “walk 5 metres” is incomplete without direction. A vector stores both size and direction.

Precise meaning

Vectors add componentwise. The magnitude of (x,y) is √(x²+y²). a·b=|a||b|cosθ; zero for nonzero vectors means perpendicular.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

start and end → components → operation → magnitude → direction/dot-product check

Worked example

For a=(3,4) and b=(−1,2), a+b=(2,6), |a|=5, and a·b=5.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Finding the length of (3,4) as 3+4=7 is wrong; perpendicular components require Pythagoras, giving 5.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. For a=(2,−3,6) and b=(1,4,−2), find sum, difference, |a|, and dot product.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The checkpoint combines proof, scale, measurement, circles, trigonometry, and vectors.

Course contents