Class 12 RD Sharma Solutions – Chapter 24 Scalar or Dot Product – Exercise 24.1 | Set 3

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Question 33. Find the angle between the two vectors $\vec{a}$ and $\vec{b}$ , if

(i) $|\vec{a}|$ =√3, $|\vec{b}|$ = 2 and $\vec{a}.\vec{b}$ = √6

Solution:

We know, $\vec{a}\vec{b} = |\vec{a}||\vec{b}|cos\theta$

⇒ √6 = 2√3 cos θ

⇒ cos θ = 1/√2

⇒ θ = cos^-1(1/√2)

⇒ θ = π/4

(ii) $|\vec{a}|$ = 3, $|\vec{b}|$ = 3 and $\vec{a}.\vec{b}$ = 1

Solution:

We know, $\vec{a}\vec{b} = |\vec{a}||\vec{b}|cos\theta$

⇒ 1 = 3×3 cos θ

⇒ cos θ = 1/9

⇒ θ = cos^-1(1/9)

Question 34. Express the vector $\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b}=3\hat{i}+\hat{k}$ and other is perpendicular to $\vec{b}$

Solution:

Given, $\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}$

$\vec{b}=3\hat{i}+\hat{k}$

Let the two vectors be $\vec{a_1}, \vec{a_2}$

Now, $\vec{a}=\vec{a_1}+\vec{a_2}$ ….(1)

Assuming $\vec{a_1}$ is parallel to $\vec{b}$

Then, $\vec{a_1}=λ\vec{b}$ ……(2)

$\vec{a_2}$ is perpendicular to $\vec{b}$

Then, $\vec{a_2}.\vec{b}=0$ ……(3)

From eq(1)

$\vec{a_1}=λ\vec{b}$

⇒ $\vec{a_2} = \vec{a}-λ\vec{b}$

⇒ $\vec{a_2} =5\hat{i}-2\hat{j}+5\hat{k} -λ(3\hat{i}+\hat{k})$

⇒ $\vec{a_2} =(5-3λ)\hat{i}-2\hat{j}+(5-λ)\hat{k}$

From eq(3)

$\vec{a_2}.\vec{b}=0$

⇒ $((5-3λ)\hat{i}-2\hat{j}+(5-λ)\hat{k})(3\hat{i}+\hat{k})$

⇒ (5-3λ)3+(5-λ)=0

⇒ 15-9λ+5-λ=0

⇒ -10λ = -20

⇒ λ=2

From eq(2)

$\vec{a_1}=6\hat{i}+2\hat{k}$

$\vec{a_2}=-\hat{i}-2\hat{j}+3\hat{k}$

$\vec{a}=(6\hat{i}+2\hat{k})+(-\hat{i}-2\hat{j}+3\hat{k})$

Question 35. If $\vec{a}$ and $\vec{b}$ are two vectors of the same magnitude inclined at an angle of 30° such that $\vec{a}.\vec{b}$ = 3, find $|\vec{a}|, |\vec{b}|.$

Solution:

Given that two vectors of the same magnitude inclined at an angle of 30°, and $\vec{a}.\vec{b} = 3$

To find $|\vec{a}|, |\vec{b}|$

We know, $\vec{a}\vec{b} = |\vec{a}||\vec{b}|cos\theta$

⇒ 3 = $|\vec{a}||\vec{a}|cos θ$

⇒ 3 = $|\vec{a}|^2 cos 30°$

⇒ 3 = $|\vec{a}|^2$ (√3/2)

⇒ $|\vec{a}|^2$ = 6/√3

⇒ $|\vec{a}|=|\vec{b}|=\sqrt{2\sqrt{3}}$

Question 36. Express $2\hat{i}-\hat{j}+3\hat{k}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{i}+4\hat{j}-2\hat{k}.$

Solution:

Assuming $\vec{a}=2\hat{i}-\hat{j}+3\hat{k}$

$\vec{b}=2\hat{i}+4\hat{j}-2\hat{k}$

Let the two vectors be $\vec{a_1}, \vec{a_2}$

Now, $\vec{a}=\vec{a_1}+\vec{a_2}$

or $\vec{a_2}=\vec{a}-\vec{a_1}$ ….(1)

Assuming $\vec{a_1}$ is parallel to $\vec{b}$

then, $\vec{a_1}=λ\vec{b}$ …(2)

$\vec{a_2}$ is perpendicular to $\vec{b}$

then, $\vec{a_2}.\vec{b}=0$ ……(3)

Putting eq(2) in eq(1), we get

$\vec{a_1}=λ\vec{b}$

⇒ $\vec{a_2} = \vec{a}-λ\vec{b}$

⇒ $\vec{a_2} =2\hat{i}-\hat{j}+3\hat{k} -λ(2\hat{i}+4\vec{j}-2\hat{k})$

⇒ $\vec{a_2} =(2-2λ)\hat{i}-(1+4λ)\hat{j}+(3+2λ)\hat{k}$

From eq(3)

$\vec{a_2}.\vec{b}=0$

⇒ $((2-2λ)\hat{i}-(1+4λ)\hat{j}+(3+2λ)\hat{k})(2\hat{i}+4\hat{j}-2\hat{k})=0$

⇒ (2 – 2λ)2 – (1 + 4λ)4 – (3 + 2λ)2 = 0

⇒ 4 – 4λ – 4 – 16λ – 6 – 4λ = 0

⇒ 24λ = -6

⇒ λ = -6/24

From eq(2)

$\vec{a_1}=-1/4(2\hat{i}+4\hat{j}-2\hat{k})$

$\vec{a_1}=-1/2\hat{i}-1\hat{j}+1/2\hat{k}$

$\vec{a_2}=(2+1/2)\hat{i}-0\hat{j}+(3-1/2)\hat{k}$

$\vec{a_2}=5/2\hat{i}+5/2\hat{k}$

$\vec{a}=\vec{a_1}+\vec{a_2}$

$\vec{a}=(-1/2\hat{i}-1\hat{j}+1/2\hat{k})+(5/2\hat{i}+5/2\hat{k})$

Question 37. Decompose the vector $6\hat{i}-3\hat{j}-6\hat{k}$ into vectors which are parallel and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}.$

Solution:

Let $\vec{a}=6\hat{i}-3\hat{j}-6\hat{k}$ and $\vec{m}=\hat{i}+\hat{j}+\hat{k}$

Let $\vec{b}$ be a vector parallel to $\hat{i}+\hat{j}+\hat{k}.$

Therefore, $\vec{b} =λ(\hat{i}+\hat{j}+\hat{k})$

$\vec{a}$ to be decomposed into two vectors

$\vec{a}=\vec{b}+\vec{c}$

⇒ $\vec{c}=\vec{a}-\vec{b}$

⇒ $\vec{c}=(6-λ)\hat{i}+(-3-λ)\hat{j}+(-6-λ)\vec{k}$

Now, $\vec{c}$ is perpendicular to $\vec{m}$

or $\vec{c}.\vec{m}=0$

⇒ $((6-λ)\hat{i}+(-3-λ)\hat{j}+(-6-λ)\vec{k})(\hat{i}+\hat{j}+\hat{k})=0$

⇒ 6 – λ – 3 – λ – 6 – λ = 0

⇒ λ = -1

Therefore, the required vectors are $\vec{b} =-\hat{i}-\hat{j}-\hat{k}$ and $\vec{c}=7\hat{i}-2\hat{j}-5\vec{k}$

Question 38. Let $\vec{a}=5\hat{i}-\hat{j}+7\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+λ\hat{k}$ . Find λ such that $\vec{a}+\vec{b}$ is orthogonal to $\vec{a}-\vec{b}$

Solution:

Given, $\vec{a}=5\hat{i}-\hat{j}+7\hat{k}$

$\vec{b}=\hat{i}-\hat{j}+λ\hat{k}$

According to question

$(\vec{a}+\vec{b})(\vec{a}-\vec{b})=0$

⇒ $|\vec{a}|^2-|\vec{b}|^2=0$

⇒ $|\vec{a}|^2=|\vec{b}|^2$

⇒ $\sqrt{5^2+(-1)^2+7^2}=\sqrt{1^2+(-1)^2+λ^2}$

⇒ 25 + 1 + 49 = 1 + 1 + λ²

⇒ λ²= 73

⇒ λ = √73

Question 39. If $\vec{a}.\vec{a}=0$ and $\vec{a}.\vec{b}=0$ , what can you conclude about the vector $\vec{b}$ ?

Solution:

Given, $\vec{a}.\vec{a}=0$ , $\vec{a}.\vec{b}=0$

$|\vec{a}| = 0$

Now, $\vec{a}.\vec{b}=0$

We conclude that $\vec{a}=0$ or $\vec{b}=0$ or θ = 90°

Thus, $\vec{b}$ can be any arbitrary vector.

Question 40. If $\vec{c}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ , then prove that it is perpendicular to both $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$

Solution:

Given $\vec{c}$ is perpendicular to both $\vec{a}$ and $\vec{b}$

$\vec{c}.\vec{a}=0$ ….(1)

$\vec{c}.\vec{b}=0$ ….(2)

To prove $\vec{c}.(\vec{a}+\vec{b})=0$ and $\vec{c}.(\vec{a}-\vec{b})=0$

Now, $\vec{c}.(\vec{a}+\vec{b})$

⇒ $\vec{c}.\vec{a}+\vec{c}.\vec{b}=0$ [From eq(1) and (2)]

Again, $\vec{c}.(\vec{a}-\vec{b})$

⇒ $\vec{c}.\vec{a}-\vec{c}.\vec{b}=0$ [From eq(1) and (2)]

Hence Proved

Question 41. If $|\vec{a}|= a$ and $|\vec{b}|= b$ , prove that $(\frac{\vec{a}}{a^2}-\frac{\vec{b}}{b^2})^2= (\frac{\vec{a}-\vec{b}}{ab})^2$

Solution:

Given, $|\vec{a}| = a$ and $|\vec{b}| = b,$

To prove

$(\frac{\vec{a}}{a^2}-\frac{\vec{b}}{b^2})^2= (\frac{\vec{a}-\vec{b}}{ab})^2.$

Taking LHS

$(\frac{\vec{a}}{a^2}-\frac{\vec{b}}{b^2})^2$

= $\frac{\vec{a}.\vec{a}}{a^4}+\frac{\vec{b}.\vec{b}}{b^4}-2\frac{\vec{a}.\vec{b}}{a^2b^2}$

= $\frac{a^2}{a^4}+\frac{b^2}{b^4}-2\frac{\vec{a}.\vec{b}}{a^2b^2}$

= $\frac{1}{a^2}+\frac{1}{b^2}-2\frac{\vec{a}.\vec{b}}{a^2b^2}$

Taking RHS

$(\frac{\vec{a}-\vec{b}}{ab})^2 \vec{d}.\vec{a}=0$

= $(\frac{a^2+b^2-2\vec{a}\vec{b}}{a^2b^2})$

= $\frac{1}{a^2}+\frac{1}{b^2}-2\frac{\vec{a}.\vec{b}}{a^2b^2}$

LHS = RHS

Hence Proved

Question 42. If $\vec{a},\vec{b},\vec{c}$ are three non- coplanar vectors such that $\vec{d}.\vec{a}=\vec{d}.\vec{b}=\vec{d}.\vec{c} =0$ then show that $\vec{d}$ is the null vector.

Solution:

Given that $\vec{d}.\vec{a}=0$

So either $\vec{d} = 0$ or $\vec{d}⊥\vec{a}=0$

Similarly, $\vec{d}.\vec{b} = 0$

Either $\vec{d}= 0$ or $\vec{d}⊥\vec{b}=0$

Also, $\vec{d}.\vec{c} =0$

So $\vec{d} = 0$ or $\vec{d}⊥\vec{c}=0$

But $\vec{d}$ can’t be perpendicular to $\vec{a},\vec{b}$ and $\vec{c}$ because $\vec{a},\vec{b},\vec{c}$ are non-coplanar.

So $\vec{d}$ = 0 or $\vec{d}$ is a null vector

Question 43. If a vector $\vec{a}$ is perpendicular to two non- collinear vectors $\vec{b}$ and $\vec{c}$ , then is $\vec{a}$ perpendicular to every vector in the plane of $\vec{b}$ and $\vec{c}$

Solution:

Given that $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$

$\vec{a}.\vec{b}=0 , \vec{a}.\vec{c}=0$

Let $\vec{r}$ be any vector in the plane of $\vec{b}$ and $\vec{c}$ and $\vec{r}$ is the linear combination of $\vec{b}$ and $\vec{c}$

$\vec{r} =x\vec{b}+y\vec{c}$ [x, y are scalars]

Now $, \vec{a}.\vec{r}$

⇒ $\vec{a}.\vec{r} =\vec{a}(x\vec{b}+y\vec{c})$

⇒ $\vec{a}.\vec{r} =x(\vec{a}.\vec{b})+y(\vec{a}.\vec{c})$

⇒ $\vec{a}.\vec{r} =x0+y0$

⇒ $\vec{a}.\vec{r} = 0$

Therefore, $\vec{a}$ is perpendicular to $\vec{r}$ i.e. $\vec{a}$ is perpendicular to every vector.

Question 44. If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ , how that the angle θ between the vectors $\vec{b}$ and $\vec{c}$ is given by cos θ = $\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}$

Solution:

Given that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$

⇒ $\vec{a}=-(\vec{b}+\vec{c})$

⇒ $(\vec{a})^2=(\vec{b}+\vec{c})^2$

⇒ $\vec{a}.\vec{a}=(\vec{b}+\vec{c})(\vec{b}+\vec{c})$

⇒ $|\vec{a}|^2=|\vec{b}|^2+|\vec{b}||\vec{c}|+|\vec{b}||\vec{c}|+|\vec{c}|^2$

⇒ $|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2=2|\vec{b}||\vec{c}|$

⇒ $|\vec{b}||\vec{c}|=(|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2)/2$

⇒ cos θ = $\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}.$

Question 45. Let $\vec{u},\vec{v}$ and $\vec{w}$ be vector such $\vec{u}+\vec{v}+\vec{w}=\vec{0}$ . $|\vec{u}|$ = 3, $|\vec{v}|$ = 4 and $|\vec{w}|$ = 5, then find $\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u}$

Solution:

Given that $\vec{u},\vec{v}$ and $\vec{w}$ are vectors such that $\vec{u}+\vec{v}+\vec{w}=\vec{0}$ . $|\vec{u}|$ = 3, $|\vec{v}|$ = 4 and $|\vec{w}|$ =5,

To find $\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u}$

Taking

$\vec{u}+\vec{v}+\vec{w}=\vec{0}$

Squaring on both side, we get

⇒ $(\vec{u}+\vec{v}+\vec{w})^2=\vec{0}$

⇒ $|\vec{u}|^2+|\vec{v}|^2+|\vec{w}|^2+2(\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u})=0$

⇒ $2(\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u})=-(|\vec{u}|^2+|\vec{v}|^2+|\vec{w}|^2)$

⇒ $2(\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u})=-(3^2+4^2+5^2)$

⇒ $2(\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u})=-50$

Therefore, $\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u}=-25$

Question 46. Let $\vec{a}=x^2\hat{i}+2\hat{j}-2\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k},$ and $\vec{c}=x^2\hat{i}+5\hat{j}-4\hat{k}$ be three vectors. Find the values of x for which the angle between $\vec{a}$ and $\vec{b}$ is acute and the angle between $\vec{b}$ and $\vec{c}$ is obtuse.

Solution:

Given $\vec{a}=x^2\hat{i}+2\hat{j}-2\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}, \vec{c}=x^2\hat{i}+5\hat{j}-4\hat{k}$

Case I: When angle between $\vec{a}$ and $\vec{b}$ is acute:-

$\vec{a}.\vec{b}$ >0

⇒ $(x^2\hat{i}+2\hat{j}-2\hat{k})(\hat{i}-\hat{j}+\hat{k})>0$

⇒ x²– 2 – 2 > 0

⇒ x²> 4

x ∈ (2, -2)

Case II: When angle between $\vec{b}$ and $\vec{c}$ is obtuse:-

$\vec{b}.\vec{c}<0$

⇒ $(\hat{i}-\hat{j}+\hat{k})(x^2\hat{i}+5\hat{j}-4\hat{k})<0$

⇒ x²– 5 – 4 < 0

⇒ x²< 9

x ∈ (3, -3)

Therefore, x ∈ (-3, -2)∪(2, 3)

Question 47. Find the value of x and y if the vectors $\vec{a}=3\hat{i}+x\hat{j}-\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+y\hat{k}$ are mutually perpendicular vectors of equal magnitude.

Solution:

Given $\vec{a}=3\hat{i}+x\hat{j}-\hat{k}, \vec{b}=2\hat{i}+\hat{j}+y\hat{k}$ are mutually perpendicular vectors of equal magnitude.

$|\vec{a}|^2=|\vec{b}|^2$

⇒ 3²+ x²+ (-1)² = 2²+ 1²+ y²

⇒ x²+10 = y²+5

⇒ x²– y²+ 5 = 0 ….(1)

Now, $\vec{a}.\vec{b} = 0$

⇒ 6 + x – y = 0

⇒ y = x + 6 …..(2)

From eq(1)

x²– (x + 6)²+ 5 = 0

⇒ x² – (x² + 36 – 12x) + 5 = 0

⇒ -12x – 31 = 0

⇒ x = -31/12

Now, y = -31/12 + 6

y = 41/12

Question 48. If $\vec{a}$ and $\vec{b}$ are two non-coplanar unit vectors such that $|\vec{a}+\vec{b}| =√3$ , find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}).$

Solution:

Given that $\vec{a}$ and $\vec{b}$ are two non-coplanar unit vectors such that $|\vec{a}+\vec{b}| =√3$

To find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}).$

Now,

$|\vec{a}+\vec{b}|^2 =3$

$⇒|\vec{a}|^2+|\vec{b}|^2+2\vec{a}.\vec{b}=3$

$⇒ 1+1+2\vec{a}.\vec{b}=3$

$⇒ \vec{a}.\vec{b}=1/2$

Now, $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b})$

= $6|\vec{a}|^2-13\vec{a}.\vec{b}-5|\vec{b}|^2$

= 6 – 13(1/2) – 5

= 1 – 13/2

= -11/2

Question 49. If $\vec{a},\vec{b}$ are two vectors such that | $\vec{a}+\vec{b}$ | = $|\vec{b}|$ , then prove that $\vec{a}+2\vec{b}$ is perpendicular to $\vec{a}.$

Solution:

To prove

$(\vec{a}+2\vec{b})\vec{a}= 0$

Now,

$\vec{a}+\vec{b} = \vec{b}$

Squaring on both side, we get

$|\vec{a}+\vec{b}|^2=|\vec{b}|^2$

⇒ $(\vec{a}+\vec{b})(\vec{a}+\vec{b}) = \vec{b}.\vec{b}$

⇒ $\vec{a}\vec{a}+\vec{a}\vec{b}+\vec{b}\vec{a}+\vec{b}\vec{b} = \vec{b}.\vec{b}$

⇒ $\vec{a}\vec{a}+2\vec{a}\vec{b} = 0$

⇒ $\vec{a}(\vec{a}+2\vec{b}) = 0$

Therefore, $\vec{a}+2\vec{b}$ is perpendicular to $\vec{a}$

My Personal Notes

Last Updated : 28 Mar, 2021

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Class 12 RD Sharma Solutions – Chapter 24 Scalar or Dot Product – Exercise 24.1 | Set 3

Question 33. Find the angle between the two vectors $\vec{a}$ and $\vec{b}$ , if

(i) $|\vec{a}|$ =√3, $|\vec{b}|$ = 2 and $\vec{a}.\vec{b}$ = √6

(ii) $|\vec{a}|$ = 3, $|\vec{b}|$ = 3 and $\vec{a}.\vec{b}$ = 1

Question 34. Express the vector $\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b}=3\hat{i}+\hat{k}$ and other is perpendicular to $\vec{b}$

Question 35. If $\vec{a}$ and $\vec{b}$ are two vectors of the same magnitude inclined at an angle of 30° such that $\vec{a}.\vec{b}$ = 3, find $|\vec{a}|, |\vec{b}|.$

Question 36. Express $2\hat{i}-\hat{j}+3\hat{k}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{i}+4\hat{j}-2\hat{k}.$

Question 37. Decompose the vector $6\hat{i}-3\hat{j}-6\hat{k}$ into vectors which are parallel and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}.$

Question 38. Let $\vec{a}=5\hat{i}-\hat{j}+7\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+λ\hat{k}$ . Find λ such that $\vec{a}+\vec{b}$ is orthogonal to $\vec{a}-\vec{b}$

Question 39. If $\vec{a}.\vec{a}=0$ and $\vec{a}.\vec{b}=0$ , what can you conclude about the vector $\vec{b}$ ?

Question 40. If $\vec{c}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ , then prove that it is perpendicular to both $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$

Question 41. If $|\vec{a}|= a$ and $|\vec{b}|= b$ , prove that $(\frac{\vec{a}}{a^2}-\frac{\vec{b}}{b^2})^2= (\frac{\vec{a}-\vec{b}}{ab})^2$

Question 42. If $\vec{a},\vec{b},\vec{c}$ are three non- coplanar vectors such that $\vec{d}.\vec{a}=\vec{d}.\vec{b}=\vec{d}.\vec{c} =0$ then show that $\vec{d}$ is the null vector.

Question 43. If a vector $\vec{a}$ is perpendicular to two non- collinear vectors $\vec{b}$ and $\vec{c}$ , then is $\vec{a}$ perpendicular to every vector in the plane of $\vec{b}$ and $\vec{c}$

Question 44. If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ , how that the angle θ between the vectors $\vec{b}$ and $\vec{c}$ is given by cos θ = $\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}$

Question 45. Let $\vec{u},\vec{v}$ and $\vec{w}$ be vector such $\vec{u}+\vec{v}+\vec{w}=\vec{0}$ . $|\vec{u}|$ = 3, $|\vec{v}|$ = 4 and $|\vec{w}|$ = 5, then find $\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u}$

Question 46. Let $\vec{a}=x^2\hat{i}+2\hat{j}-2\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k},$ and $\vec{c}=x^2\hat{i}+5\hat{j}-4\hat{k}$ be three vectors. Find the values of x for which the angle between $\vec{a}$ and $\vec{b}$ is acute and the angle between $\vec{b}$ and $\vec{c}$ is obtuse.

Question 47. Find the value of x and y if the vectors $\vec{a}=3\hat{i}+x\hat{j}-\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+y\hat{k}$ are mutually perpendicular vectors of equal magnitude.

Question 48. If $\vec{a}$ and $\vec{b}$ are two non-coplanar unit vectors such that $|\vec{a}+\vec{b}| =√3$ , find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}).$

Question 49. If $\vec{a},\vec{b}$ are two vectors such that | $\vec{a}+\vec{b}$ | = $|\vec{b}|$ , then prove that $\vec{a}+2\vec{b}$ is perpendicular to $\vec{a}.$

CBSE Class 12 Computer Science

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Class 12 RD Sharma Solutions – Chapter 24 Scalar or Dot Product – Exercise 24.1 | Set 3

Question 33. Find the angle between the two vectors and , if

(i) =√3, = 2 and = √6

(ii) = 3, = 3 and = 1

Question 34. Express the vector as the sum of two vectors such that one is parallel to the vector and other is perpendicular to

Question 35. If and are two vectors of the same magnitude inclined at an angle of 30° such that = 3, find

Question 36. Express as the sum of a vector parallel and a vector perpendicular to

Question 37. Decompose the vector into vectors which are parallel and perpendicular to the vector

Question 38. Let and . Find λ such that is orthogonal to

Question 39. If and , what can you conclude about the vector ?

Question 40. If is perpendicular to both and , then prove that it is perpendicular to both and

Question 41. If and , prove that

Question 42. If are three non- coplanar vectors such that then show that is the null vector.

Question 43. If a vector is perpendicular to two non- collinear vectors and , then is perpendicular to every vector in the plane of and

Question 44. If , how that the angle θ between the vectors and is given by cos θ =

Question 45. Let and be vector such . = 3, = 4 and = 5, then find

Question 46. Let and be three vectors. Find the values of x for which the angle between and is acute and the angle between and is obtuse.

Question 47. Find the value of x and y if the vectorsand are mutually perpendicular vectors of equal magnitude.

Question 48. If and are two non-coplanar unit vectors such that , find

Question 49. If are two vectors such that || = , then prove that is perpendicular to

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CBSE Class 12 Computer Science

GATE CS & IT 2024

Complete Machine Learning & Data Science Program

Question 33. Find the angle between the two vectors $\vec{a}$ and $\vec{b}$ , if

(i) $|\vec{a}|$ =√3, $|\vec{b}|$ = 2 and $\vec{a}.\vec{b}$ = √6

(ii) $|\vec{a}|$ = 3, $|\vec{b}|$ = 3 and $\vec{a}.\vec{b}$ = 1

Question 34. Express the vector $\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b}=3\hat{i}+\hat{k}$ and other is perpendicular to $\vec{b}$

Question 35. If $\vec{a}$ and $\vec{b}$ are two vectors of the same magnitude inclined at an angle of 30° such that $\vec{a}.\vec{b}$ = 3, find $|\vec{a}|, |\vec{b}|.$

Question 36. Express $2\hat{i}-\hat{j}+3\hat{k}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{i}+4\hat{j}-2\hat{k}.$

Question 37. Decompose the vector $6\hat{i}-3\hat{j}-6\hat{k}$ into vectors which are parallel and perpendicular to the vector $\hat{i}+\hat{j}+\hat{k}.$

Question 38. Let $\vec{a}=5\hat{i}-\hat{j}+7\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+λ\hat{k}$ . Find λ such that $\vec{a}+\vec{b}$ is orthogonal to $\vec{a}-\vec{b}$

Question 39. If $\vec{a}.\vec{a}=0$ and $\vec{a}.\vec{b}=0$ , what can you conclude about the vector $\vec{b}$ ?

Question 40. If $\vec{c}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ , then prove that it is perpendicular to both $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$

Question 41. If $|\vec{a}|= a$ and $|\vec{b}|= b$ , prove that $(\frac{\vec{a}}{a^2}-\frac{\vec{b}}{b^2})^2= (\frac{\vec{a}-\vec{b}}{ab})^2$

Question 42. If $\vec{a},\vec{b},\vec{c}$ are three non- coplanar vectors such that $\vec{d}.\vec{a}=\vec{d}.\vec{b}=\vec{d}.\vec{c} =0$ then show that $\vec{d}$ is the null vector.

Question 43. If a vector $\vec{a}$ is perpendicular to two non- collinear vectors $\vec{b}$ and $\vec{c}$ , then is $\vec{a}$ perpendicular to every vector in the plane of $\vec{b}$ and $\vec{c}$

Question 44. If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ , how that the angle θ between the vectors $\vec{b}$ and $\vec{c}$ is given by cos θ = $\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}$

Question 45. Let $\vec{u},\vec{v}$ and $\vec{w}$ be vector such $\vec{u}+\vec{v}+\vec{w}=\vec{0}$ . $|\vec{u}|$ = 3, $|\vec{v}|$ = 4 and $|\vec{w}|$ = 5, then find $\vec{u}.\vec{v}+\vec{v}.\vec{w}+\vec{w}.\vec{u}$

Question 46. Let $\vec{a}=x^2\hat{i}+2\hat{j}-2\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k},$ and $\vec{c}=x^2\hat{i}+5\hat{j}-4\hat{k}$ be three vectors. Find the values of x for which the angle between $\vec{a}$ and $\vec{b}$ is acute and the angle between $\vec{b}$ and $\vec{c}$ is obtuse.

Question 47. Find the value of x and y if the vectors $\vec{a}=3\hat{i}+x\hat{j}-\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+y\hat{k}$ are mutually perpendicular vectors of equal magnitude.

Question 48. If $\vec{a}$ and $\vec{b}$ are two non-coplanar unit vectors such that $|\vec{a}+\vec{b}| =√3$ , find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}).$

Question 49. If $\vec{a},\vec{b}$ are two vectors such that | $\vec{a}+\vec{b}$ | = $|\vec{b}|$ , then prove that $\vec{a}+2\vec{b}$ is perpendicular to $\vec{a}.$