## EXERCISE 4.3

#### Page No 447:

#### Question 1:

Find the angle between two vectorsandwith magnitudesand 2, respectively having.

#### Answer:

It is given that,

Now, we know that.

Hence, the angle between the given vectors andis.

#### Question 2:

Find the angle between the vectors

#### Answer:

The given vectors are.

Also, we know that.

#### Question 3:

Find the projection of the vectoron the vector.

#### Answer:

Letand.

Now, projection of vectoronis given by,

Hence, the projection of vector onis 0.

#### Question 4:

Find the projection of the vectoron the vector.

#### Answer:

Letand.

Now, projection of vectoronis given by,

#### Question 5:

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

#### Answer:

Thus, each of the given three vectors is a unit vector.

Hence, the given three vectors are mutually perpendicular to each other.

#### Page No 448:

#### Question 6:

Findand, if.

#### Answer:

#### Question 7:

Evaluate the product.

#### Answer:

#### Question 8:

Find the magnitude of two vectors, having the same magnitude and such that the angle between them is 60° and their scalar product is.

#### Answer:

Let *θ* be the angle between the vectors

It is given that

We know that.

#### Question 9:

Find, if for a unit vector.

#### Answer:

#### Question 10:

Ifare such thatis perpendicular to, then find the value of *λ*.

#### Answer:

Hence, the required value of *λ* is 8.

#### Question 11:

Show that is perpendicular to, for any two nonzero vectors

#### Answer:

Hence, andare perpendicular to each other.

#### Question 12:

If, then what can be concluded about the vector?

#### Answer:

It is given that.

Hence, vectorsatisfyingcan be any vector.

#### Question 13:

If are unit vectors such that , find the value of .

#### Answer:

It is given that .

From (1), (2) and (3),

#### Question 14:

If either vector, then. But the converse need not be true. Justify your answer with an example.

#### Answer:

We now observe that:

Hence, the converse of the given statement need not be true.

#### Question 15:

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectorsand]

#### Answer:

The vertices of ΔABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2).

Also, it is given that ∠ABC is the angle between the vectorsand.

Now, it is known that:

.

#### Question 16:

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

#### Answer:

The given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Hence, the given points A, B, and C are collinear.

#### Question 17:

Show that the vectorsform the vertices of a right angled triangle.

#### Answer:

Let vectors be position vectors of points A, B, and C respectively.

Now, vectorsrepresent the sides of ΔABC.

Hence, ΔABC is a right-angled triangle.

#### Question 18:

Ifis a nonzero vector of magnitude ‘*a*’ and λ a nonzero scalar, then *λ*is unit vector if

(A) λ = 1 (B) λ = –1 (C)

(D)

#### Answer:

Vectoris a unit vector if.

Hence, vectoris a unit vector if.

The correct answer is D.