Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# If a vector $\overrightarrow{A}$ makes angles $\alpha,\beta$ and $\gamma$ with $X$,$Y$ and $Z$ axes respectively then $sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=$\begin{align} & A.0 \\ & B.1 \\ & C.2 \\ & D.3 \\ \end{align}

Last updated date: 19th Jul 2024
Total views: 383.4k
Views today: 10.83k
Verified
383.4k+ views
Hint: We know that a vector has both length or magnitude and direction. The angle the vector makes with the axis give the direction or the orientation of the vector on any given space. Generally, a vector is said to make $\alpha,\beta$ and $\gamma$ with $X$,$Y$ and $Z$ axes respectively.

Given that the vector $\overrightarrow{A}$ makes angles $\alpha,\beta$ and $\gamma$with $X$,$Y$ and $Z$ axes respectively. This implies that the unit vector $\hat{a}$, also known as the directional vector is as shown in the figure. Then we can express the vector $\overrightarrow{A}$ as $\vec A=a_{x}\hat i+a_{y}\hat j+a_{z}\hat k$, and the unit vector $\hat{a}$ is given as, $\hat a=\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}$

We can then express, the x,y,z components of the vectors with respect to the angles as the following:
$cos\alpha=\dfrac{a_{x}\hat i}{\hat a}$
$cos\beta=\dfrac{a_{y}\hat j}{\hat a}$
$cos\gamma=\dfrac{a_{z}\hat k}{\hat a}$
Squaring and adding, we get, $cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=\dfrac{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}{\left(\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}\right)^{2}}=1$
Thus, $cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1$
But we need $sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma$
We know from trigonometry identities, that $sin^{2}\theta+cos^{2}\theta=1$
Then, we can write, $cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1-sin^{2}\alpha+1-sin^{2}\beta+1-sin^{2}\gamma=3-(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma)$
Then,$3-(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma)=1$
Or,$sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=3-1=2$
Thus, $sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=2$
Hence C.$2$ is the answer.

Note: When we say a $\vec{AB}$, we mean that $A$ is carried to $B$ in a particular direction in the space. Also magnitude denoted as $|\vec{AB}|$ gives the distance between the points $A$ and $B$. these vectors are generally represented on the coordinate systems.