Answer
Verified
412.2k+ 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.
Complete step by step answer:
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.
Additional information:
We know that two vectors can be added, subtracted, and multiplied. We also know that the algebraic laws such as commutativity, associativity and distributivity are valid to certain degree and not always applicable. If a scalar and a vector is multiplied, it is said to be scalar multiplication or dot product. If a vector and another vector are multiplied, then it is said to be cross product or vector multiplication.
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.
Complete step by step answer:
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.
Additional information:
We know that two vectors can be added, subtracted, and multiplied. We also know that the algebraic laws such as commutativity, associativity and distributivity are valid to certain degree and not always applicable. If a scalar and a vector is multiplied, it is said to be scalar multiplication or dot product. If a vector and another vector are multiplied, then it is said to be cross product or vector multiplication.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE