Question
What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$
✓
Answer
Given:
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{0}$
$\Rightarrow\vec{\text{a}}=0$
$\vec{\text{b}}=0$
$\therefore\vec{\text{a}}||\vec{\text{b}}$
Also,
$\vec{\text{a}}.\vec{\text{b}}=0$
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$
$\Rightarrow\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0}$ or, $\vec{\text{a}}\perp\vec{\text{b}}$
But $\vec{\text{a}}$ cannot be both perpendicular as well as parallel to $\vec{\text{b}}.$
$\therefore|\vec{\text{a}}|=0$
$\big|\vec{\text{b}}\big|=0$
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{0}$
$\Rightarrow\vec{\text{a}}=0$
$\vec{\text{b}}=0$
$\therefore\vec{\text{a}}||\vec{\text{b}}$
Also,
$\vec{\text{a}}.\vec{\text{b}}=0$
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta=0$
$\Rightarrow\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0}$ or, $\vec{\text{a}}\perp\vec{\text{b}}$
But $\vec{\text{a}}$ cannot be both perpendicular as well as parallel to $\vec{\text{b}}.$
$\therefore|\vec{\text{a}}|=0$
$\big|\vec{\text{b}}\big|=0$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the vector equation of the plane passing through the pointA(-2, 3, 5) and parallel to
→vectors $4 \hat{i}+3 \hat{k}$ and $\hat{i}+\hat{j}$
Find the vector equation of a line passing through (2, -1, 1) and parallel to the line whose equations are $\frac{\text{x}-3}{2}=\frac{\text{y}+1}{7}=\frac{\text{z}-2}{-3}.$
→Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
If $\big(\sin^{-1}\text{x}\big)^2+\big(\sin^{-1}\text{y}\big)^2+\big(\sin^{-1}\text{z}\big)^2=\frac{3}{4}\pi^2,$ find the value of $x^2 + y^2 + z^2$
→Find $\frac{d y}{d x}$ if : $x^5+x y^3+x^2 y+y^4=4$
→Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
Evaluate the following integrals:
$\int\tan^\frac{3}{2}\text{x}\sec^2\text{x dx}$
→$\int\tan^\frac{3}{2}\text{x}\sec^2\text{x dx}$
Verify $y=\log x+c$ is the solution of differential equation $x \frac{ d ^2 y}{ d x^2}+\frac{ d y}{ d x}=0$
→A man of height 2 meters walks at a uniform speed of 6 km/hr away from a lamp post of 6 meters high. Find the rate at which the length of the shadow is increasing.
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3 - 6x^2 - 36x + 2$
→$f(x) = x^3 - 6x^2 - 36x + 2$