## Rajasthan Board RBSE Class 12 Maths Chapter 13 Vector Ex 13.5

Question 1.

Find the value of

Solution:

Question 2.

Prove that

Solution:

Question 3.

Evaluate the formula

Solution:

Question 4.

For any vector \(\overrightarrow { a } \), prove that:

Solution:

We know that

Question 5.

Prove that

Solution:

Question 6.

Prove that \(\overrightarrow { a } \),\(\overrightarrow { b } \),\(\overrightarrow { c } \) are coplanar if \(\overrightarrow { a } \) × \(\overrightarrow { b } \), \(\overrightarrow { b } \) × \(\overrightarrow { c } \), \(\overrightarrow { c } \) × [/latex], \(\overrightarrow { a } \) are coplanar

Solution:

Question 7.

Prove that

Solution:

Question 8.

If magnitude of two \(\overrightarrow { a } \) and \(\overrightarrow { b } \) are √3 and 2 respectively \(\overrightarrow { a } \) and. \(\overrightarrow { b } \) = √6, then find the angle between \(\overrightarrow { a } \) and \(\overrightarrow { b } \)

Solution:

Question 9.

Find the angle between the vectors

and

Solution:

Question 10.

Find the projection of vector

Solution:

Question 11.

Find projection vector on vector

on vector

Solution:

Question 12.

Find the value of

Solution:

Question 13.

Find the magnitude of two vectors \(\overrightarrow { a } \) and \(\overrightarrow { b } \), if their magnitude are equal and angle between them is 60° and their scalar product is \(\frac { 1 }{ 2 } \)

Solution:

According to question

Question 14.

If for a vector \(\overrightarrow { a } \), (\(\overrightarrow { x } \) – \(\overrightarrow { a } \) ).( \(\overrightarrow { x } \) + \(\overrightarrow { a } \) ) = 12, then | \(\overrightarrow { x } \) |.

Solution:

According to question,

Question 15.

such that \(\overrightarrow { a } \) + λ \(\overrightarrow { b } \) is perpendicular on \(\overrightarrow { c } \), then find the value of λ.

Solution:

According to question \(\overrightarrow { a } \) + λ \(\overrightarrow { b } \) is a perpendicular to \(\overrightarrow { c } \)

Question 16.

If vertices \(\overrightarrow { a } \), \(\overrightarrow { b } \), \(\overrightarrow { c } \) are such that

\(\overrightarrow { a } \) + \(\overrightarrow { b } \) + \(\overrightarrow { c } \) = \(\overrightarrow { 0 } \)

then, find the value of \(\overrightarrow { a } \) . \(\overrightarrow { b } \) + \(\overrightarrow { b } \) . \(\overrightarrow { c } \) + \(\overrightarrow { c } \) . \(\overrightarrow { a } \)

Solution:

According to question

Question 17.

If vectors/1, B, C of triangle ABC are (1, 2,3), (-1,0, 0,), (0, 1, 2) respectively, then find ∠ABC.

Solution:

Let O be the origin.