If with reference to the right handed system of mutually perpendi… - Vidyadip

Question

If with reference to the right handed system of mutually perpendicular unit vectors $\hat{i}, \hat{j}$ and $\hat{k}, \vec{\alpha}=3 \hat{i}-\hat{j.}$ $\vec{\beta}=2 \hat{i}+\hat{j}-3 \hat{k.}$ then express $\vec{\beta}$ in the form $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, where $\vec{\beta}_1$ is $\|$ to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.

✓

Answer

Let $\vec{\beta}_1=\lambda \vec{\alpha}\left[\because \vec{\beta}_1 \| t o \vec{\alpha}\right]$
$\vec{\beta}_1=\lambda(3 \hat{i}-\hat{j})$
$=3 \lambda \hat{i}-\lambda \hat{j}$
$\vec{\beta}_2=\vec{\beta}-\vec{\beta}_1$
$=(2-3 \lambda) \hat{i}+(1+\lambda) \hat{j}-3 \hat{k}$
$\vec{\alpha} \cdot \vec{\beta}_2=0\left[\because \vec{\beta}_2 \perp \vec{\alpha}\right]$
$3(2-3 \lambda)-(1+\lambda)=0$
$\lambda=\frac{1}{2}$
$\vec{\beta}_1=\frac{3}{2} \hat{i}-\frac{1}{2} \hat{j}$
$\vec{\beta}_2=\frac{1}{2} \hat{i}+\frac{3}{2} \hat{j}-3 \hat{k}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Model Paper 3→Model Paper 3 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the area of the triangle with vertices at the points:
$(2, 7), (1, 1)$ and $(10, 8)$

Using elementary transformation, find the inverse of each of the matrices, $\begin{bmatrix}3 & 10 \\2 & 7 \end{bmatrix}$

If $\text{y}=\sec^{-1}\Big(\frac{\text{x}+1}{\text{x}-1}\Big)+\sin^{-1}\Big(\frac{\text{x}-1}{\text{x}+1}\Big),\text{x}>0.$ Find $\frac{\text{dy}}{\text{dx}}.$

Evaluate the following:
$\tan^{-1}1+\cos^{-1}\Big(-\frac{1}{2}\Big)+\sin^{-1}\Big(-\frac{1}{2}\Big)$

Let $A = {1, 2, 3},$ and let $R_1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}$. Find whether or not the relations $R_1$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

Show that $\text{f}(\text{x})=\log\sin\text{x}$ is increasing on $\Big(0,\frac{\pi}{2}\Big)$ and decreasing on $\Big(\frac{\pi}{2},\pi\Big).$

Without expanding the determinant, prove that $\begin{vmatrix}a&a^2&bc\\b&b^2&ca\\c&c^2&ab\end{vmatrix}=\begin{vmatrix}1&a^2&a^3\\1&b^2&b^3\\1&b^2&b^3\end{vmatrix}.$

Three relation $R_3$ is defined in set $A=\{a, b, c\}$ as follows:
$R_3=\{(b, c)\}$
Find whether or not the relation $R_3$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

If two sides of a triangle be represented by the vectors $\hat{i}+2 \hat{j}+2 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$, then prove that the area of the triangle is $\frac{5}{2} \sqrt{5}$ square units.

If for a binomial distribution P(X = 1) = P(X = 2) = α, write P(X = 4) in terma of α.