Download App

Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space5 Marks
Question
Find the vector equation of a line which is parallel to the vector $2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and which passes through the point (5, -2, 4). Also, reduce it to cartesian from.
✓

Answer

We know that, vector equation of line passing through a fixed point $\vec{\text{a}}$ and paralel to vector $\vec{\text{b}}$ is $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}},$ where $\lambda$ is scalar Here, $\vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{a}}=5\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}$ So, rquation of required line is $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}$ $\vec{\text{r}}=\big(5\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}\big)$ Put $\vec{\text{r}}=\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}},$ so $\big(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}}\big)=(5+2\lambda)\hat{\text{i}}+(-2-\lambda)\hat{\text{j}}+(4+3\lambda)\hat{\text{k}}$ Comparing the cofficients of $\hat{\text{i}},\hat{\text{j}},\hat{\text{k}}$ so $\text{x}=5+2\lambda,\text{y}=-2-\lambda,\text{z}=4+3\lambda$ $\Rightarrow\frac{\text{x}-5}{2}=\lambda,\frac{\text{y}+2}{-0}=\lambda,\frac{\text{z}-4}{3}=\lambda$ Cortesian form of equation of the line is,$\frac{\text{x}-5}{5}=\frac{\text{y}+2}{-0}=\frac{\text{z}-4}{3}$

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 Straight line in space→Straight line in space — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\sin\text{x}\sin2\text{x}\sin3\text{x}\sin4\text{x}$
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(\theta+\sin\theta)$ and $\text{y}=\text{a}(1-\cos\theta)$
→
Integrate the function in Exercise:
$(\text{x}^2+1)\text{log}\ \text{x}$
→
Solve the following differential equations:
$\text{y}\sqrt{1+\text{x}^2}+\text{x}\sqrt{1+\text{y}^2}\ \frac{\text{dy}}{\text{dx}}=0$
→
Find the local maximum and local minima, of the function $\text{f(x)} = \sin x - \cos x, 0< x < 2\pi.$ Also find the local maximum and local minimum values.
→
Show that the function $\text{f(x)}\begin{cases}\text{x}^\text{m}\sin(\frac{1}{\text{x}}), &\text{x}\neq0 \\0 ,& \text{x}=0\end{cases}$
Neirher continuous but not diffierentiable, if $\text{m}\leq0$
→
Using matrix method, solve the following system of equations:
$\frac{2}{\text{x}}+\frac{3}{\text{y}}+\frac{10}{\text{z}}=4,\frac{4}{\text{x}}-\frac{6}{\text{x}}+\frac{5}{\text{z}}=1,\frac{6}{\text{x}}+\frac{9}{\text{y}}-\frac{20}{\text{z}}=2; \text{ x,y,z,}\neq0$
→
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
→
If $\Big(\sin^{-1}\text{x}\Big)^2+\Big(\cos^{-1}\text{x}\Big)^2=\frac{175\pi^2}{36},$ find x.
→
AB is the diameter of a circle and C is any point on the circle. Show that the area of triangle ABC is maximum, when it is an isosceles triangle.
→