With reference to the right handed system of mutually perpendicul… - Vidyadip

Question

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

✓

Answer

Let, $\overrightarrow{\beta_1}=\lambda \vec{\alpha}$ is a scalar,
i.e., $\overrightarrow{\beta_1}=3 \lambda \vec{i}-\lambda \vec{j}$
Now, $\overrightarrow{\beta_2}=\vec{\beta}-\overrightarrow{\beta_1}=(2-3 \lambda) \hat{i}+(1+\lambda) \hat{j}-3 \hat{k}$
Now, since $\overrightarrow{\beta_2}$ is to be perpendicular $\vec{\alpha}$
we should have $\vec{\alpha} \cdot \overrightarrow{\beta_2}=0$.
i.e., $3(2-3 \lambda)-(1+\lambda)=0$
OR $\lambda=\frac{1}{2}$.
Therefore, $\overrightarrow{\beta_1}=\frac{3}{2} \widehat{i}-\frac{1}{2} \widehat{j}$ and $\overrightarrow{\beta_2}=\frac{1}{2} \widehat{i}+\frac{3}{2} \widehat{j}-3 \widehat{k}$

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" from JUNE 2024→JUNE 2024 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evalute the following integrals:
$\int\big\{1+\tan\text{x}\tan(\text{x}+\theta)\big\}\text{dx}$

Evaluate the following integrals:
$\int\sec^6\text{x }\tan\text{x}\text{ dx}$

Find the domain of $\text{f(x)}=2\cos^{-1}2\text{x}=\sin^{-1}\text{x}.$

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^x sin x.

Represent the following families of curves by forming the corresponding differential equation:
$(\text{x}-\text{a})^2-\text{y}^2=1$

Evaluate the following integrals:
$\int_{4}^\limits{12}\text{x}(\text{x}-4)^{\frac{1}{3}}\text{dx}$

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibits 4.

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x² + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate.

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$