Download App

Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors5 Marks
Question
The position vectors of points A, B and C are $\lambda\hat{\text{i}}+3\hat{\text{j}},12\hat{\text{i}}+\mu\hat{\text{j}}\text{ and }11\hat{\text{i}}-3\hat{\text{j}}$ respectively. If C divides the line segment joining A and B in the ratio 3:1, find the value of $\lambda\text{ and }\mu$
✓

Answer

The position vectors of points A, B and C are $\lambda\hat{\text{i}}+3\hat{\text{j}},12\hat{\text{i}}+\mu\hat{\text{j}}\text{ and }11\hat{\text{i}}-3\hat{\text{j}}$, respectively.
It is given that, C divides the line segment joining A and B in the ratio 3 : 1.
$11\hat{\text{i}}-3\hat{\text{j}}=\frac{3\times\big(12\hat{\text{i}}+\mu\hat{\text{j}}\big)+1\times\big(\lambda\hat{\text{i}}+3\hat{\text{j}}\big)}{3+1}$
$\Rightarrow11\hat{\text{i}}-3\hat{\text{j}}=\frac{(36+\lambda)\hat{\text{i}}+(3\mu+3)\hat{\text{j}}}{4}$
$\Rightarrow44\hat{\text{i}}-12\hat{\text{j}}=(36+\lambda)\hat{\text{i}}+(3\mu+3)\hat{\text{j}}$
Equating the corresponding components, we get
$36+\lambda=44$
$\Rightarrow\lambda=44-36=8$
and
$3\mu+3=-12$
$\Rightarrow3\mu=-12-3$
$\Rightarrow\mu=-5$
Thus the values of $\lambda\text{ and }\mu$ are 8 and -5, respectively.

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 "5 Marks Questions" from Algebra of Vectors→Algebra of Vectors — 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 equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=\frac{\text{x}^3}{4-\text{x}}\text{ at }(2,-2)$
→
Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5+4\sin\text{x}}\text{ dx}$
→
Evaluate the following integrals:
$\int\limits^{1}_0\big|\text{x}\sin\pi\text{x}\big|\text{dx}$
→
$\text{If }\overrightarrow{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\hat{\text{7k}};\text{ and }\overrightarrow{b}=\hat{\text{5i}}-\hat{\text{j}}+\lambda\hat{\text{k}},$ then find the value of $\lambda,$ so that $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ are perpendicular vectors.
→
Find the points o local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be:
$\text{f}(\text{x})=\sin2\text{x}-{\text{x}},-\frac{\pi}{2}\leq\text{x}\leq\frac{\pi}{2}$
→
Draw a rough sketch of the region {$(x, y) : y^2 < 3x, 3x^2 + < 16$} and find the area by the region using mwthod of integration.
→
If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-1}{\text{x}-1},& \text{for }\text{ x}\neq1 \\2,&\text{for }\text{ x}=1\end{cases}$ Find whether f (x) is continuous at x = 1.
→
Show that the following system of linear equation is inconsistent:
$2x + 3y = 5$
$6x + 9y = 10$
→
A and B are two independent events. The probability that A and B occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Find the probability of occurrence of two events.
→
Find the angle between the vectors with direction ratios proportional to 1, -2, 1 and 4, 3, 2.
→