Relative to the origin O , the position vectors of the points P and Q are given by O P → = 5 k - 3 7 k + 9 , O Q → = k k + 2 - 1 where k is a constant. Find the values of k for which O P → and O Q → . are perpendicular to one another.

Given that the vectors O P → = 5 k - 3 7 k + 9 & O Q → = k k + 2 - 1 are perpendicular We have to find the value of the constant k As we know that, a → · b → = a b cos θ ⇒ cos θ = a → · b → a b Since, the given vectors are perpendicular So, O P → · O Q → = 0 Therefore, 5 k k + - 3 k + 2 + 7 k + 9 - 1 = 0 ⇒ 5 k 2 - 3 k - 6 - 7 k - 9 = 0 ⇒ 5 k 2 - 10 k - 15 = 0 ⇒ k 2 - 2 k - 3 = 0 ⇒ k + 1 k - 3 = 0 ⇒ k = - 1 , 3 Hence, the required value of k = - 1 , 3 .

Relative to the origin O , the position vectors of the points P and Q are given by O P → = 5 k - 3 7 k + 9 , O Q → = k k + 2 - 1 where k is a constant. Given that k = 2 , find the angle, θ , between O P → and O Q → .

Given that the vectors O P → = 5 k - 3 7 k + 9 & O Q → = k k + 2 - 1 are perpendicular We have to find the value of the constant k If k = 2 , then O P → = 10 - 3 23 & O Q → = 2 4 - 1 As we know that, a → · b → = a b cos θ ⇒ cos θ = a → · b → a b Since, the given vectors are perpendicular So, cos θ = 10 × 2 + - 3 × 4 + 23 × - 1 10 2 + - 3 2 + 23 2 2 2 + 4 2 + - 1 2 ⇒ cos θ = - 15 638 21 ⇒ θ = cos - 1 - 15 638 21 ⇒ θ = 97 . 4 ° Hence, the required angle is 97 . 4 ° .

The position vector a is such that a = 4 i - 8 j + k . Find a · j and hence find the angle that a makes with the positive y -axis.

Given that: The position vector a is such that a = 4 i - 8 j + k . We have to find a · j and hence find the angle that a makes with the positive y -axis. Let the angle between them be θ As we know that, a → · b → = a b cos θ ⇒ cos θ = a → · b → a b So, cos θ = 4 × 0 - 8 × 1 + 1 × 0 4 2 + - 8 2 + 1 2 0 2 + 1 2 + 0 2 ⇒ cos θ = - 8 81 1 = - 8 9 ⇒ θ = cos - 1 - 8 9 = 152 . 733 ° ≈ 152 . 7 ° So, the required angle is 152 . 7 ° .

Explain why a · b · c has no meaning.

We have to explain why a · b · c has no meaning. a · b is a scalar and the dot product is a product of two vectors a · b · c is dot product is a product of scalar and vector has no meaning.

HARD

AS and A Level

IMPORTANT

Earn 100

Find which of the following pairs of position vectors are perpendicular to one another.

For any position vectors that are not perpendicular, find the acute angle between them.

$c = (\begin{matrix} 5 \\ 12 \\ 13 \end{matrix}), d = (\begin{array}{r} - 3 \\ - 1 \\ 3 \end{array})$

Important Questions on Vectors

MEDIUM

AS and A Level

IMPORTANT

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

Find which of the following pairs of position vectors are perpendicular to one another.

For any position vectors that are hot perpendicular, find the acute angle between them.

$e = (\begin{array}{r} 4 \\ - 9 \\ - 2 \end{array}), f = (\begin{array}{r} - 4 \\ - 2 \\ 1 \end{array})$

HARD

AS and A Level

IMPORTANT

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

Relative to the origin $O,$ the position vectors of the points $A$ and $B$ are given by $\vec{O A} = (\begin{array}{r} - 5 \\ 0 \\ 3 \end{array})$ and $\vec{O B} = (\begin{array}{l} 1 \\ 7 \\ 2 \end{array}) .$

Find angle $B O A .$

HARD

AS and A Level

IMPORTANT

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

Given that the vectors

6 i - 2 j + 5 k

and

a i + 4 j - 2 k

are perpendicular, find the value of the constant

a .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9C>Q 4

Relative to the origin $O,$ the position vectors of the points $P$ and $Q$ are given by $\vec{O P} = (\begin{matrix} 5 k \\ - 3 \\ 7 k + 9 \end{matrix}), \vec{O Q} = (\begin{matrix} k \\ k + 2 \\ - 1 \end{matrix})$ where $k$ is a constant.

Find the values of $k$ for which $\vec{O P}$ and $\vec{O Q} .$ are perpendicular to one another.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9C>Q 4

Given that $k = 2,$ find the angle, $θ,$ between $\vec{O P}$ and $\vec{O Q} .$

HARD

AS and A Level

IMPORTANT

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

The diagram shows a cuboid $O A B C D E F G$ in which $O C = 6 cm,$ $O A = 4 cm$ and $O D = 3 cm .$ Unit vectors $i, j$ and $k$ are parallel to $O C, O A$ and $O D$ respectively. The point $M$ lies on $F B$ such that $F M = 1 cm, N$ is the midpoint of $O C$ and $P$ is the centre of the rectangular face $D E F G .$

Question Image

Using a scalar product, find angle $N P M .$

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9C>Q 6

The position vector

a

is such that

a = 4 i - 8 j + k .

Find

a \cdot j

and hence find the angle that a makes with the positive

y

-axis.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 9 - Vectors>EXERCISE 9C>Q 7

Explain why

a \cdot b \cdot c

has no meaning.

Find which of the following pairs of position vectors are perpendicular to one another. For any position vectors that are not perpendicular, find the acute angle between them. c=51213, d=-3-13

Important Questions on Vectors

Learn and Prepare for any exam you want !

Find which of the following pairs of position vectors are perpendicular to one another.

For any position vectors that are not perpendicular, find the acute angle between them.

$c = (\begin{matrix} 5 \\ 12 \\ 13 \end{matrix}), d = (\begin{array}{r} - 3 \\ - 1 \\ 3 \end{array})$