Solve both parts and justify the conditions used. Part A: Explain why parallel vectors cannot form a basis for the plane. Part B: Give a unit vector in the same direction.

01 / Transfer#Your turn

\begin{gathered}\text{A: }\mathbf e_2=2\mathbf e_1\\\text{B: }\mathbf a=(24,32)\end{gathered}

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use a nonparallel basis or Cartesian coordinates and solve component equations.

Hint 2

B: Separate magnitude, direction and position; compare vectors by displacement.

Worked solution

Part A reasoning
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Calculate or simplify this relation.
$a\mathbf e_1+b\mathbf e_2=(a+2b)\mathbf e_1$
They cannot represent a vector outside that line.
Part B reasoning
Separate magnitude, direction and position; compare vectors by displacement.
Calculate or simplify this relation.
$\frac{\mathbf a}{|\mathbf a|}=(3/5,4/5)$
Normalizing a zero vector would be undefined.

A: All their linear combinations stay on one line. B: (3/5,4/5).

Checks and common pitfalls: They cannot represent a vector outside that line. Normalizing a zero vector would be undefined.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Plane vectors and applications: mixed assessment