1.10 Vectors

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Question 47

With respect to a fixed origin $O$ , a research drone is monitoring a straight-line path $l_1$ given by the equation

\mathbf{r} = \begin{pmatrix} -3 \\ 2 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}

where $μ\mu$ is a scalar parameter.

The point $A$ represents a specific checkpoint on $l_1$ . Given that $∣OA→∣=70|\overrightarrow{OA}| = \sqrt{70}$ .

Show that at $A$ , the parameter $μ\mu$ satisfies

9\mu^2 - 28\mu - 32 = 0

[4]

(i) Show that one possible position vector for $A$ is $5i+6j−3k5\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$ . (ii) Find the other possible position vector for $A$ .

[3]

The path $l_2$ is parallel to $l_1$ and passes through the origin $O$ . Given that:

$OA→=5i+6j−3k\overrightarrow{OA} = 5\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$
Point $B$ lies on $l_2$ where $∣OB→∣=15|\overrightarrow{OB}| = 15$

Find the area of triangle $O A B$ , giving your answer to one decimal place.

[5]

1.10 Vectors

Question 47

With respect to a fixed origin $O$ , a research drone is monitoring a straight-line path $l_1$ given by the equation

\mathbf{r} = \begin{pmatrix} -3 \\ 2 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}

where $μ\mu$ is a scalar parameter.

The point $A$ represents a specific checkpoint on $l_1$ . Given that $∣OA→∣=70|\overrightarrow{OA}| = \sqrt{70}$ .

Show that at $A$ , the parameter $μ\mu$ satisfies

9\mu^2 - 28\mu - 32 = 0

[4]

(i) Show that one possible position vector for $A$ is $5i+6j−3k5\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$ . (ii) Find the other possible position vector for $A$ .

[3]

The path $l_2$ is parallel to $l_1$ and passes through the origin $O$ . Given that:

$OA→=5i+6j−3k\overrightarrow{OA} = 5\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}$
Point $B$ lies on $l_2$ where $∣OB→∣=15|\overrightarrow{OB}| = 15$

Find the area of triangle $O A B$ , giving your answer to one decimal place.

[5]

1.10 Vectors

Question 47

Markscheme

1.10 Vectors

Question 47

Markscheme

1.10 Vectors

Question 47

1.10 Vectors Questions

1.10 Vectors

Question 47

1.10 Vectors Questions