Find the angles made by the vectors displaystyle{A}={5}{i}-{2}{j}+{3}{k} with the axes give a full correct answer

Question

Find the angles made by the vectors

A = 5 i - 2 j + 3 k

with the axes give a full correct answer

Answer 1

First note that the cosines of the angles a vector makes with the Cartesian coordinate axes are called the direction cosines.
Now let the given vector is

A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

Let the vector A makes andgle

α, β and γ

with cordinate axes x, y and z, respectively.
Also, let l, m and n are the direction cosine along x-axis, y-axis and z-axis, respectively.
Then we have

l = \cos α = \frac{A_{x}}{\frac{{(A_{x}^{2} + A_{y}^{2} + A_{z}^{2})}^{1}}{2}}

m = \cos β = \frac{A_{y}}{\frac{{(A_{x}^{2} + A_{y}^{2} + A_{z}^{2})}^{1}}{2}}

n = \cos γ = \frac{A_{z}}{\frac{{(A_{x}^{2} + A_{y}^{2} + A_{z}^{2})}^{1}}{2}}

It is given that

A_{x} = 5, A_{y} = - 2, A_{z} = 3

Thus we have

\frac{{(A_{x}^{2} + A_{y}^{2} + A_{z}^{2})}^{1}}{2} = \sqrt{5^{2} + {(- 2)}^{2} + 3^{2}} = \sqrt{38}

Thus angle with x-axis is

α = \cos^{- 1} (\frac{5}{\sqrt{38}}) = {35.78}^{\circ}

The angle with y-axis is

β = \cos^{- 1} (\frac{- 1}{\sqrt{38}}) = {108.93}^{\circ}

The angle with z-axis is

γ = \cos^{- 1} (\frac{3}{\sqrt{38}}) = {60.88}^{\circ}

Find the angles made by the vectors displaystyle{A}={5}{i}-{2}{j}+{3}{k} with the axes give a full correct answer

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