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Scalar Or Dot Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSScalar Or Dot Product5 Marks
Question
Find the angles which the vector $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\sqrt{2}\hat{\text{k}}$ makes with the coordinate axes.
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Answer

Let $\theta_{1}$ be the angle between $\vec{\text{a}}$ and x-axis.
$\big|\vec{\text{a}}\big|=\sqrt{(1)^2+(-1)^2+(\sqrt{2})^2}=\sqrt{4}=2$
$\vec{\text{b}}=\hat{\text{i}}$ (Because $\hat{\text{i}}$ is the unit vector along x-axis)
$\big|\vec{\text{b}}\big|=\sqrt{(1)^2}=\sqrt{1}=1$
$\vec{\text{a}}.\vec{\text{b}}=1+0+0=1$
$\cos\theta_{1}=\frac{\vec{\text{a}} .\vec{\text{b}}}{\big|\vec{a}\big|\big|\vec{b}\big|}=\frac{1}{(2)(1)}=\frac{1}{2}$
$\Rightarrow\theta_{1}=\cos^{-1}\big(\frac{1}{2}\big)=\frac{\pi}{3}$
Let $\theta_{2}$ be the angle between $\vec{\text{a}}$ and y-axis.
$\big|\vec{\text{a}}\big|=\sqrt{(1)^2+(-1)^2+(\sqrt{2})^2}=\sqrt{4}=2$
$\vec{\text{b}}=\hat{\text{j}}$(Because $\hat{\text{j}}$ is the unit vector along y-axis)
$\big|\vec{\text{b}}\big|=\sqrt{(1)^2}=\sqrt{1}=1$
$\vec{\text{a}}.\vec{\text{b}}=0-1+0=-1$
$\cos\theta_{2}=\frac{\vec{\text{a}} .\vec{\text{b}}}{\big|\vec{a}\big|\big|\vec{b}\big|}=\frac{-1}{(2)(1)}=\frac{-1}{2}$
$\Rightarrow\theta_{2}=\cos^{-1}\big(\frac{-1}{2}\big)=\frac{2\pi}{3}$
Let $\theta_{3}$ be the angle between $\vec{\text{a}}$ and z-axis.
$\big|\vec{\text{a}}\big|=\sqrt{(1)^2+(-1)^2+(\sqrt{2})^2}=\sqrt{4}=2$
$\vec{\text{b}}=\hat{\text{k}}$ (Because $\hat{\text{k}}$ is the unit vector along z-axis)
$\big|\vec{\text{b}}\big|=\sqrt{(1)^2}=\sqrt{1}=1$
$\vec{\text{a}}.\vec{\text{b}}=0+0+\sqrt{2}=\sqrt{2}$
$\cos\theta=\frac{\vec{\text{a}} .\vec{\text{b}}}{\big|\vec{a}\big|\big|\vec{b}\big|}=\frac{\sqrt{2}}{(2)(1)}=\frac{1}{\sqrt2}$
$\Rightarrow\theta=\cos^{-1}\big(\frac{1}{\sqrt2}\big)=\frac{\pi}{4}$

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