[Maximum number: 6]

Relative to an origin O, the position vectors of points A and B are given by

\overrightarrow{O A}=\left(\begin{array}{r} 3 \\ -6 \\ p \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 2 \\ -6 \\ -7 \end{array}\right)

and angle $A O B=90^{\circ}$ .

(a)

Find the value of p.

[ 2 ]

(b)

The point C is such that $\overrightarrow{O C}=\frac{2}{3} \overrightarrow{O A}$ .

Find the unit vector in the direction of $\overrightarrow{B C}$ .

[ 4 ]

[Maximum number: 4]

Relative to an origin O, the position vectors of points A and B are given by

\overrightarrow{O A}=2 \mathbf{i}-5 \mathbf{j}-2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=4 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k}

The point C is such that $\overrightarrow{A B}=\overrightarrow{B C}$ . Find the unit vector in the direction of $\overrightarrow{O C}$ .

[Maximum number: 5]

Relative to an origin O, the position vectors of the points A, B and C are given by

\overrightarrow{O A}=\left(\begin{array}{r} 2 \\ -1 \\ 4 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 4 \\ 2 \\ -2 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 1 \\ 3 \\ p \end{array}\right)

Find

(a)

the unit vector in the direction of $\overrightarrow{A B}$ ,

[ 3 ]

(b)

the value of the constant p for which angle $B O C=90^{\circ}$ .

[ 2 ]

[Maximum number: 6]

The diagram shows a pyramid O A B C D in which the vertical edge O D is 3 units in length. The point E is the centre of the horizontal rectangular base O A B C. The sides O A and A B have lengths of 6 units and 4 units respectively. The unit vectors i, j and k are parallel to $\overrightarrow{O A}, \overrightarrow{O C}$ and $\overrightarrow{O D}$ respectively.

(a)

Express each of the vectors $\overrightarrow{D B}$ and $\overrightarrow{D E}$ in terms of i, j and k.

[ 2 ]

(b)

Use a scalar product to find angle B D E.

[ 4 ]

[Maximum number: 6]

Relative to an origin O, the position vectors of points A and B are given by

\overrightarrow{O A}=\left(\begin{array}{l} 5 \\ 1 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ 4 \\ -3 \end{array}\right)

The point P lies on A B and is such that $\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$ .

(a)

Find the position vector of P.

[ 3 ]

(b)

Find the distance O P.

[ 1 ]

(c)

Determine whether O P is perpendicular to A B. Justify your answer.

[ 2 ]

[Maximum number: 6]

The diagram shows a pyramid O A B C in which the edge O C is vertical. The horizontal base O A B is a triangle, right-angled at O, and D is the mid-point of A B. The edges O A, O B and O C have lengths of 8 units, 6 units and 10 units respectively. The unit vectors i, j and k are parallel to $\overrightarrow{O A}, \overrightarrow{O B}$ and $\overrightarrow{O C}$ respectively.

(a)

Express each of the vectors $\overrightarrow{O D}$ and $\overrightarrow{C D}$ in terms of i, j and k.

[ 2 ]

(b)

Use a scalar product to find angle O D C.

[ 4 ]

[Maximum number: 6]

The diagram shows a prism A B C D P Q R S with a horizontal square base A P S D with sides of length 6 cm . The cross-section A B C D is a trapezium and is such that the vertical edges A B and D C are of lengths 5 cm and 2 cm respectively. Unit vectors i, j and k are parallel to A D, A P and A B respectively.

(a)

Express each of the vectors $\overrightarrow{C P}$ and $\overrightarrow{C Q}$ in terms of i, j and k.

[ 2 ]

(b)

Use a scalar product to calculate angle P C Q.

[ 4 ]

[Maximum number: 3]

Relative to an origin O, the position vectors of points A and B are given by

\overrightarrow{O A}=5 \mathbf{i}+\mathbf{j}+2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=2 \mathbf{i}+7 \mathbf{j}+p \mathbf{k}

where p is a constant.

(a)

Find the value of p for which angle A O B is $90^{\circ}$ .

[ 3 ]

(b)

In the case where p=4, find the vector which has magnitude 28 and is in the same direction as $\overrightarrow{A B}$ .

[Maximum number: 7]

Two vectors, u and v, are such that

\mathbf{u}=\left(\begin{array}{l} q \\ 2 \\ 6 \end{array}\right) \quad \text { and } \quad \mathbf{v}=\left(\begin{array}{c} 8 \\ q-1 \\ q^{2}-7 \end{array}\right)

where q is a constant.

(a)

Find the values of q for which u is perpendicular to v.

[ 3 ]

(b)

Find the angle between u and v when q=0.

[ 4 ]

[Maximum number: 7]

In the diagram, O A B C D E F G is a rectangular block in which $O A=O D=6 \mathrm{~cm}$ and $A B=12 \mathrm{~cm}$ . The unit vectors i, j and k are parallel to $\overrightarrow{O A}, \overrightarrow{O C}$ and $\overrightarrow{O D}$ respectively. The point P is the mid-point of D G, Q is the centre of the square face C B F G and R lies on A B such that $A R=4 \mathrm{~cm}$ .

(a)

Express each of the vectors $\overrightarrow{P Q}$ and $\overrightarrow{R Q}$ in terms of i, j and k.

[ 3 ]

(b)

Use a scalar product to find angle R Q P.

[ 4 ]