Question

If a vector is equal to two icap - 2 j cap - k cap and b vector is equal to icap + 2 j cap then find the angle between a vector and b vector find the projection of resultant vector of a vector and b vector on x axis

Answer 1

Answer:

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Answer 2

The angle between vector A and vector B = (π -  $cos^{-1} \frac{2}{3\sqrt{5} }$ ) and the projection of the resultant vector on the x-axis = 2.

Given:

We are given two vectors

A = 2i - 2j - k

B = i + 2j

To Find:

i) The angle between A and B.

ii) The projection of the resultant vector on the x-axis

Solution:

We are given two vectors A and B such that

A = 2i - 2j - k

B = i + 2j

i) We need to find the angle between A and B.  The angle between two vectors A and B, denoted by θ, is the angle between their tails and is given using the formula:

A.B = |A||B|cosθ

A.B = (2i - 2j - k)(i + 2j) = 2 - 4 = -2

|A||B| = $(\sqrt{2^{2} + (-2)^{2} + 1^{2} } )(\sqrt{1^{2} + 2^{2} } )$ = $(\sqrt{9})(\sqrt{5} )$ = 3$\sqrt{5}$

⇒ -2 =  3$\sqrt{5}$cosθ

⇒ cosθ = -2/(3$\sqrt{5}$)

⇒ θ = $cos^{-1} \frac{-2}{3\sqrt{5} }$ = π -  $cos^{-1} \frac{2}{3\sqrt{5} }$  

ii) We need to find the projection of the resultant vector on the x-axis.

The resultant vector R is the sum of the two vectors A and B.

∴ R = A + B

R = (2i - 2j - k) + (i + 2j) = 3i - k

The projection of resultant vector on x -axis  = R.i = (2i - 2j - k).i = 2

∴ The angle between vector A and vector B = (π -  $cos^{-1} \frac{2}{3\sqrt{5} }$ ) and the projection of the resultant vector on the x-axis = 2.

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