Question

Sometimes a vector is described in terms of its components by using parenthesis notation. $\overrightarrow{A}$ is written as $\overrightarrow{A}=\left(A_x,A_y,A_z\right)$. What angle does the vector $\overrightarrow{A}=\left(1,1,1\right)$ make with the X,Y,Z axes?

Answer 1

$\begin{array}{c}\overrightarrow{a}=i+j+k\\ \hat{a}=\frac{i+j+k}{\sqrt{1^2+1^2+1^2}}=\frac{i+j+k}{\sqrt{3}}\end{array}$

angle from x-axis lie on z plane

$\begin{array}{c}j+k\\ \cos \theta =\frac{a.b}{|a||b|}\\ =\frac{(i+j+k)(j+k)}{\sqrt{{\left(1\right)}^2+{\left(1\right)}^2+{\left(1\right)}^2}\sqrt{{\left(1\right)}^2+{\left(1\right)}^2}}\\ =\frac{1+1}{\sqrt{2}\sqrt{3}}=\frac{2}{\sqrt{2}\sqrt{3}}\\ \theta ={\cos }^{-1}\frac{2}{\sqrt{2}\sqrt{3}}=54.73\end{array}$