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Learn Exercise 10.4 with Free Lessons & Tips

If either  or, then. Is the converse true? Justify your answer with an example.

Take any parallel non-zero vectors so that.

It can now be observed that:

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

Comments

Find, if and.

We have,

and

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Find a unit vector perpendicular to each of the vector and, where and.

We have,

and

Hence, the unit vector perpendicular to each of the vectors and is given by the relation,

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If a unit vector  makes an angleswith with and an acute angle θ with, then find θ and hence, the compounds of.

Let unit vector have (a1, a2, a3) components.

Since is a unit vector, .

Also, it is given that makes angleswith with , and an acute angle θ with

Then, we have:

Hence, and the components of are.

Comments

Show that

Comments

Find λ and μ if .

On comparing the corresponding components, we have:

Hence,

Comments

Given that  and. What can you conclude about the vectors?

Then,

(i) Either or, or

(ii) Either or, or

But, and cannot be perpendicular and parallel simultaneously.

Hence, or.

Comments

Let the vectors given as. Then show that 

We have,

On adding (2) and (3), we get:

Now, from (1) and (4), we have:

Hence, the given result is proved.

Comments

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and

C (1, 5, 5).

The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sidesand of ΔABC are given as:

Area of ΔABC

Hence, the area of ΔABC

Comments

Find the area of the parallelogram whose adjacent sides are determined by the vector .

The area of the parallelogram whose adjacent sides are is.

Adjacent sides are given as:

Hence, the area of the given parallelogram is.

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Let the vectors and be such that and, then is a unit vector, if the angle between and is

(A) (B) (C) (D)

It is given that.

We know that, where is a unit vector perpendicular to both and and θ is the angle between and.

Now, is a unit vector if.

Hence, is a unit vector if the angle between and is.

The correct answer is B.

Comments

Area of a rectangle having vertices A, B, C, and D with position vectors and  respectively is

(A) 

(B) 1

(C) 2

(D) 

The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

The adjacent sides and  of the given rectangle are given as:

???AB×BC???=2⇒AB→×BC→=2

Now, it is known that the area of a parallelogram whose adjacent sides are is.

Hence, the area of the given rectangle is

The correct answer is C.

Comments

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