Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

Find the angle between two vectors and with magnitudes and 2, respectively having .

Answer:

It is given that,

Now, we know that .

Hence, the angle between the given vectors  and is .

Question 2:

Find the angle between the vectors

Answer:

The given vectors are .

Also, we know that .

Question 3:

Find the projection of the vector on the vector .

Answer:

Let and .

Now, projection of vector on is given by,

Hence, the projection of vector  on is 0.

Question 4:

Find the projection of the vector on the vector .

Answer:

Let and .

Now, projection of vector on is 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.

Question 6:

Find and , 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:

If are such that is 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,  and are perpendicular to each other.

Question 12:

If , then what can be concluded about the vector ?

Answer:

It is given that .

Hence, vector satisfying can 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 vectors and ]

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 vectors and .

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 vectors form the vertices of a right angled triangle.

Answer:

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

Now, vectors represent the sides of ΔABC.

Hence, ΔABC is a right-angled triangle.

Question 18:

If is 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:

Vector is a unit vector if .

Hence, vector is a unit vector if .

The correct Answer is D.