Download App

Straight line in space — Maths STD 12 Science — Question

Maharashtra 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 "Solve the Following Question.(5 Marks)" from Straight line in space→Straight line in space — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Radium decomposes at a rate proportional to the amount present at any time. If p percent of the amount disappears in one year, what percent of the amount of radium will be left after 2 years?
→
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
→
Solve each of the following L.P.P. : Maximize z = 4x + 2y subject to 3x + y ≥ 27, x + y ≥ 21 Question is modified. Maximize z = 4x + 2y subject to 3x + y ≤ 27, x + y ≤ 21, x ≥ 0, y ≥ 0
→
A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.
→
Evaluate the following integrals:
$\int\frac{\text{x}}{3\text{x}^4-18\text{x}^2+11}\text{dx}$
→
Integrate the following w.r.t. x:

$\frac{1}{\sin x+\sin 2 x}$

→
Evaluate the following intregals:
$\int\frac{3\text{x}+1}{\sqrt{5-2\text{x}-\text{x}^2}}\text{dx}$
→
Evaluate the following integrals:$\int\limits^{\text{a}}_{-\text{a}}\frac{1}{1+\text{a}^{\text{x}}}\text{ dx},\text{ a}>0$
→
A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that.
  1. there is at least an even chance of drawing a heart.
  2. the probability of drawing a heart is greater than $\frac{3}{4}$?
→
Evaluate the following definite integrals:$\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$
→