1.13 J: Vectors

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Question 53

A research satellite's trajectory $l_1\,$ is modeled relative to a fixed origin $O\,$ by the equation

\mathbf{r} = \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}

where $\lambda\,$ is a scalar parameter. A tracking station at the origin $O\,$ detects the satellite at point $A\,$ on $l_1\,$ when its distance from $O\,$ is $\sqrt{29}\,$ units.

Show that at $A$ , the parameter $\lambda\,$ satisfies

9\lambda^2 - 8\lambda - 20 = 0

[4]

(i) Show that one possible position vector for $A\,$ is $5i+2j5\mathbf{i} + 2\mathbf{j}$ .

(ii) Find the other possible position vector for $A$ .

[3]

The signal beam $l_2\,$ from the station is parallel to $l_1\,$ and passes through $O$ . A maintenance probe $B\,$ lies on $l_2$ .

Given that:

$OA→=5i+2j\overrightarrow{OA} = 5\mathbf{i} + 2\mathbf{j}$
point $B\,$ lies on $l_2\,$ where $∣OB→∣=10|\overrightarrow{OB}| = 10$

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

[4]

1.13 J: Vectors

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Question 53

A research satellite's trajectory $l_1\,$ is modeled relative to a fixed origin $O\,$ by the equation

\mathbf{r} = \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}

where $\lambda\,$ is a scalar parameter. A tracking station at the origin $O\,$ detects the satellite at point $A\,$ on $l_1\,$ when its distance from $O\,$ is $\sqrt{29}\,$ units.

Show that at $A$ , the parameter $\lambda\,$ satisfies

9\lambda^2 - 8\lambda - 20 = 0

[4]

(i) Show that one possible position vector for $A\,$ is $5i+2j5\mathbf{i} + 2\mathbf{j}$ .

(ii) Find the other possible position vector for $A$ .

[3]

The signal beam $l_2\,$ from the station is parallel to $l_1\,$ and passes through $O$ . A maintenance probe $B\,$ lies on $l_2$ .

Given that:

$OA→=5i+2j\overrightarrow{OA} = 5\mathbf{i} + 2\mathbf{j}$
point $B\,$ lies on $l_2\,$ where $∣OB→∣=10|\overrightarrow{OB}| = 10$

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

[4]

1.13 J: Vectors

Question 53

1.13 J: Vectors Questions

1.13 J: Vectors

Question 53

1.13 J: Vectors Questions