Relative to the origin O , the position vectors of points A , B and C are given by O A → = 2 3 7 , O B → = 3 m 1 1 and O C → = 4 - m - m ( m + 1 ) It is given that A B → = O C → . Find the angle O A B .

Given that: O A → = 2 3 7 , O B → = 3 m 1 1 and O C → = 4 - m - m ( m + 1 ) We have to find the angle O A B . It is given that A B → = O C → So, 3 m - 2 1 - 3 1 - 7 = 4 - m - m m + 1 Using y - component, we get - 2 = - m ⇒ m = 2 Now, A B → = 3 2 - 2 - 2 - 6 = 4 - 2 - 6 Let the angle O A B be θ So, cos θ = 2 × 4 + 3 - 2 + 7 - 6 2 2 + 3 2 + 7 2 4 2 + - 2 2 + - 6 2 cos θ = - 40 2 14 62 θ = cos - 40 2 14 62 = 137 . 2466 ° ≈ 137 . 2 ° Hence, the required angle be 137 . 2 ° .

Relative to the origin O , the position vectors of points A , B and C are given by O A → = 2 3 7 , O B → = 3 m 1 1 and O C → = 4 - m - m ( m + 1 ) Find the vector equation of the line A C .

Given that: O A → = 2 3 7 , O B → = 3 m 1 1 and O C → = 4 - m - m ( m + 1 ) We have to find the vector equation of the line A C . It is given that A B → = O C → So, 3 m - 2 1 - 3 1 - 7 = 4 - m - m m + 1 Using y - component, we get - 2 = - m ⇒ m = 2 Now, A B → = 3 2 - 2 - 2 - 6 = 4 - 2 - 6 The vector equation of the line A C is r → = O A → + λ O C → - O A → So, the required equation r → = 2 3 7 + λ 4 - 2 - 6 - 2 3 7 \ r → = 2 3 7 + λ 2 - 5 - 13 Or, r → = 2 i + 3 j + 7 k + λ 2 i - 5 j - 13 k .

With respect to the origin O , the position vectors of the points A and B are given by O A → = - 2 0 6 and O B → = 1 - 1 4 Find whether or not the vectors O A → and O B → are perpendicular.

Given that: O A → = - 2 0 6 and O B → = 1 - 1 4 We have to find whether or not the vectors O A → and O B → are perpendicular. As we know that, If a → and b → are perpendicular, then a → · b → = 0 So, O A → · O B → = - 2 1 + 0 - 1 + 6 4 = 22 ≠ 0 Hence, O A → and O B → are not perpendicular.

With respect to the origin O , the position vectors of the points A and B are given by O A → = - 2 0 6 and O B → = 1 - 1 4 Write down the vector A B → .

Given that: O A → = - 2 0 6 and O B → = 1 - 1 4 We have to Write down the vector A B → . As we know that, The position vector of A B → = O B → - O A → So, A B → = 1 - 1 4 - - 2 0 6 = 3 - 1 - 2 Hence, A B → = 3 - 1 - 2 Or, A B → = 3 i - j - 2 k

With respect to the origin O, the position vectors of the points A and B are given by O A → = - 2 0 6 and O B → = 1 - 1 4 Find a vector equation of the line A B .

Given that: O A → = - 2 0 6 and O B → = 1 - 1 4 We have to find a vector equation of the line A B . As we know that, The position vector of A B → = O B → - O A → So, A B → = 1 - 1 4 - - 2 0 6 = 3 - 1 - 2 Now, the vector equation of A B → is r → = O A → + λ O B → - O A → So, r → = - 2 i + 6 k + λ 3 i - j - 2 k

HARD

AS and A Level

IMPORTANT

Earn 100

The point A has coordinates $(1, 0, 5)$ and the point B has coordinates $(- 1, 2, 9)$ .
Find the acute angle between the line AB and the line L with vector equation $r = i + 3 j + 4 k + μ (- i - 2 j + 3 k) .$

Important Questions on Vectors

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9E>Q 5

The point

A

has coordinates

(1, 0, 5)

and the point

B

has coordinates

(- 1, 2, 9)

. Find the point of intersection of the line

A B

and the line

L

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 1

Relative to the origin

O

, the position vectors of points

A, B

and

C

are given by

\vec{O A} = (\begin{array}{l} 2 \\ 3 \\ 7 \end{array}), \vec{O B} = (\begin{matrix} 3 m \\ 1 \\ 1 \end{matrix}) and \vec{O C} = (\begin{matrix} 4 \\ - m \\ - m (m + 1) \end{matrix})

It is given that

\vec{A B} = \vec{O C}

. Find the angle

O A B .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 1

Relative to the origin

O

, the position vectors of points

A, B

and

C

are given by

\vec{O A} = (\begin{array}{l} 2 \\ 3 \\ 7 \end{array}), \vec{O B} = (\begin{matrix} 3 m \\ 1 \\ 1 \end{matrix}) and \vec{O C} = (\begin{matrix} 4 \\ - m \\ - m (m + 1) \end{matrix})

Find the vector equation of the line

A C

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 2

With respect to the origin

O

, the position vectors of the points

A

and

B

are given by

\vec{O A} = (\begin{array}{r} - 2 \\ 0 \\ 6 \end{array}) and \vec{O B} = (\begin{array}{r} 1 \\ - 1 \\ 4 \end{array})

Find whether or not the vectors

\vec{O A}

and

\vec{O B}

are perpendicular.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 2

With respect to the origin $O$ , the position vectors of the points $A$ and $B$ are given by
$\vec{O A} = (\begin{array}{r} - 2 \\ 0 \\ 6 \end{array}) and \vec{O B} = (\begin{array}{r} 1 \\ - 1 \\ 4 \end{array})$
Write down the vector $\vec{A B} .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 2

With respect to the origin O, the position vectors of the points A and B are given by
$\vec{O A} = (\begin{array}{r} - 2 \\ 0 \\ 6 \end{array}) and \vec{O B} = (\begin{array}{r} 1 \\ - 1 \\ 4 \end{array})$

Find a vector equation of the line $A B$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 2

With respect to the origin

O

, the position vectors of the points

A

and

B

are given by

\vec{O A} = (\begin{array}{r} - 2 \\ 0 \\ 6 \end{array}) and \vec{O B} = (\begin{array}{r} 1 \\ - 1 \\ 4 \end{array})

The vector equation of the line

L

is given by

r = 7 i - 12 j + 7 k + μ (- 3 i + 10 j - 5 k)

Show that the lines

A B

and

L

intersect and find the position vector of the point of intersection.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>END-OF-CHAPTER REVIEW EXERCISE 9>Q 3

Question Image

The diagram shows a prism, $A B C D E F G H$ , with a parallelogram-shaped uniform cross-section.
The point $E$ is such that $O E$ is the height of the parallelogram. The point $M$ is such that $\vec{O M}$ is parallel to $\vec{D C}$ and N is the midpoint of $D E$ . The side OD has a length of $5$ units. The unit vectors $i, j$ and $k$ are parallel to $\vec{O A}, \vec{O M}$ and $\vec{O E}$ , respectively.
The position vectors of the points $A, E, C$ and $M$ are given by $\vec{O A} = 9 i, \vec{O M} = 15 j$ and $\vec{O E} = 12 k .$

Express the vectors $\vec{A H}$ and $\vec{N H}$ in terms of $i, j$ and $k .$

The point A has coordinates (1, 0, 5) and the point B has coordinates (-1, 2, 9). Find the acute angle between the line AB and the line L with vector equation r=i+3j+4k+μ(-i-2j+3k).

Important Questions on Vectors

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The point A has coordinates $(1, 0, 5)$ and the point B has coordinates $(- 1, 2, 9)$ .
Find the acute angle between the line AB and the line L with vector equation $r = i + 3 j + 4 k + μ (- i - 2 j + 3 k) .$