Review mastery and choose the next practice

Review mastery and choose the next practice

Choose one skill you cannot yet explain without looking. Follow its link, then attempt the named example or discussion question before revealing the working. Use the chapter overview to reopen this checklist. Completing a page and being able to use its method are different checkpoints.

Use the same two passes as the source checklist: at the end of the lecture, try these from memory; after the tutorial, revisit the ones you could not explain. The source’s two columns are blank self-assessment spaces, not completed mastery claims. Use each practice question’s existing self-check to record your confidence after attempting it.

Check the foundations from memory

For basic properties in two and three dimensions, check that you can:

1. Read and write vector notation: printed bold $\mathbf a$, handwritten underlined $\underline a$, displacement $\overrightarrow{AB}$, a coordinate column $\begin{pmatrix}x\\y\\z\end{pmatrix}$ and $x\mathbf i+y\mathbf j+z\mathbf k$.
2. Add and subtract vectors and explain their geometry.
3. Multiply a vector by a scalar and explain the change of length and direction.
4. Find the displacement from one given point to another.
5. Calculate a vector’s magnitude or length.
6. Find a unit vector, checking that the original vector is nonzero.
7. Draw a diagram that helps represent a vector problem.
8. Apply the ratio theorem to locate a point dividing a segment in a given ratio.
9. Use collinearity and/or ratio equations to solve vector geometry problems.

Check the product methods from memory

For scalar and vector products, check that you can:

1. Define and calculate a scalar (dot) product.
2. Manipulate scalar-product expressions using its laws.
3. Test perpendicularity with a scalar product and formulate the corresponding equation.
4. Calculate the angle between vectors placed with common initial points.
5. Define and calculate a vector (cross) product in three dimensions.
6. Manipulate vector-product expressions using its laws.
7. Find triangle and parallelogram areas with a vector product.
8. Choose the appropriate scalar or vector product for the length parallel or perpendicular to a given direction.

Choose practice for the skill you need

The source groups practice into eleven question types. Choose a row that matches your uncertainty, follow its lesson link and attempt the named example, then use Discussion and Review for another context. The example numbers here refer to this chapter’s current lecture examples.

Unit vectors

Start with: Example 1(iii)

Check your method: Normalise before scaling; specify orientation.

Magnitude

Start with: Example 1(ii)

Check your method: Take the square root of the sum of squared components.

Equal vectors

Start with: Discussion: Question 2

Check your method: Compare the appropriate components or directed sides.

Ratio theorem

Start with: Example 5

Check your method: Draw internal or produced-line point order before weighting endpoints.

Parallel or collinear vectors

Start with: Example 4

Check your method: Require one consistent scalar multiple and a common point for collinearity.

Dot products

Start with: Example 9

Check your method: Distinguish the scalar result from vector inputs.

Cross products

Start with: Example 17

Check your method: Preserve factor order, then check both dot products.

Geometrical interpretation

Start with: Example 15

Check your method: Distinguish a length from a parallel or residual vector.

Projection length

Start with: Example 15(i)

Check your method: Normalise the target direction and retain absolute value for length.

Angles

Start with: Example 12

Check your method: Use common-tail vectors and the requested rounding.

Triangle or parallelogram area

Start with: Example 18

Check your method: Use adjacent sides, then decide whether a factor one-half is needed.

These method choices use the reviewed current questions. The original source index is retained with its historical subpart references; some refer to earlier Review numbering, so use the chapter’s current links to find questions.