Practise questions · Salam Academy

SA-9709-P3-VE-0031 markL1 FoundationCalculatorScalar product

Find

\mathbf{a}\cdot\mathbf{b}

where

\mathbf{a}=\begin{pmatrix}3\\-1\\2\end{pmatrix}

and

\mathbf{b}=\begin{pmatrix}1\\4\\-2\end{pmatrix}

.

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SA-9709-P3-VE2-0051 markL1 FoundationCalculatorScalar product

|\mathbf{a}|=3

,

|\mathbf{b}|=4

and the angle between

\mathbf{a}

and

\mathbf{b}

is

60^\circ

. Find

\mathbf{a}\cdot\mathbf{b}

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE2-G011 markL1 FoundationCalculatorScalar product

\mathbf{a}=2\mathbf{i}+\mathbf{j}+3\mathbf{k}

and

\mathbf{b}=\mathbf{i}-4\mathbf{j}+2\mathbf{k}

. Find

\mathbf{a}\cdot\mathbf{b}

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE2-Q011 markL1 FoundationCalculatorScalar product

\mathbf{a}\cdot\mathbf{b}<0

tells you that the angle between

\mathbf{a}

and

\mathbf{b}

is

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE-Q031 markL1 FoundationCalculatorPerpendicular

For non-zero vectors,

\mathbf{a}\cdot\mathbf{b}=0

means

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SA-9709-P3-VE-0052 marksL2 StandardCalculatorPerpendicular

The vectors

\begin{pmatrix}2\\p\\-1\end{pmatrix}

and

\begin{pmatrix}3\\1\\4\end{pmatrix}

are perpendicular. Find

p

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE2-0013 marksL2 StandardCalculatorScalar product

Find the angle between

\mathbf{a}=3\mathbf{i}+4\mathbf{k}

and

\mathbf{b}=5\mathbf{j}+12\mathbf{k}

, in degrees correct to 1 decimal place.

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SA-9709-P3-VE2-0022 marksL2 StandardCalculatorPerpendicular

The vectors

\begin{pmatrix}k\\2\\-1\end{pmatrix}

and

\begin{pmatrix}3\\k\\4\end{pmatrix}

are perpendicular. Find

k

.

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SA-9709-P3-VE2-0043 marksL2 StandardCalculatorPerpendicular

A(1,0,2)

,

B(3,1,1)

and

C(4,k,3)

. Angle

ABC

is a right angle. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE2-0072 marksL2 StandardCalculatorScalar product

Find the angle between the vector

2\mathbf{i}+3\mathbf{j}+6\mathbf{k}

and the

x

-axis, in degrees correct to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE-0043 marksL3 ExamCalculatorScalar product

Find the angle between

\mathbf{p}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}

and

\mathbf{q}=2\mathbf{i}-\mathbf{j}+2\mathbf{k}

, giving your answer in degrees correct to 1 decimal place.

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SA-9709-P3-VE2-0034 marksL3 ExamCalculatorTriangles

The points

A(1,2,0)

,

B(3,3,2)

and

C(2,0,3)

form a triangle. Find angle

BAC

, in degrees correct to 1 decimal place.

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SA-9709-P3-VE2-0064 marksL3 ExamCalculatorPerpendicular

The vector

\mathbf{i}+p\mathbf{j}+q\mathbf{k}

is perpendicular to both

\mathbf{i}+\mathbf{j}+\mathbf{k}

and

\mathbf{i}-\mathbf{j}+2\mathbf{k}

. Find

p

and

q

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-VE2-0085 marksL4 ChallengingCalculatorTriangles

\overrightarrow{OA}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}

and

\overrightarrow{OB}=2\mathbf{i}-\mathbf{j}+2\mathbf{k}

. Find the exact area of triangle

OAB

.

Salam Academy Original Practice · Proposed mark scheme

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