2020HSCShort answer

Q11 (d)

3 marks

Question

Consider the two vectors $\mathbf u=-2\mathbf i-\mathbf j+3\mathbf k$ and $\mathbf v=p\mathbf i+\mathbf j+2\mathbf k$ . For what values of $p$ are $\mathbf u-\mathbf v$ and $\mathbf u+\mathbf v$ perpendicular?

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Solution

Criteria	Marks
Provides correction solution	3
Correctly obtains $(\vec u-\vec v)\cdot(\vec u+\vec v)$ , or equivalent merit<br>OR<br>Attempts to apply $(\vec u-\vec v)\cdot(\vec u+\vec v)=0$ using correct $(\vec u-\vec v)$ and $(\vec u+\vec v)$	2
Obtains $(\vec u-\vec v)$ and $(\vec u+\vec v)$ , or equivalent merit	1

Sample answer:

\vec u=-2\vec i-\vec j+3\vec k

\vec v=p\vec i+\vec j+2\vec k

\vec u-\vec v=(-2-p)\vec i-2\vec j+\vec k

\vec u+\vec v=(p-2)\vec i+5\vec k

(\vec u-\vec v)\cdot(\vec u+\vec v)=0

-(p+2)(p-2)+5=0

p^2-4=5

p=\pm3

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Syllabus alignment

Vectors

Further operations with three-dimensional vectorsUse vector magnitude, unit vectors and scalar products to solve problems and prove geometric results.

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