Question
If $\vec{\text{c}}$ is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}},$ then prove that it is perpendicular to both $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}.$
✓
Answer
Given that $\vec{\text{c}}$ is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}}.$
$\Rightarrow\vec{\text{c}}.\vec{\text{a}}=0$ and $\vec{\text{c}}.\vec{\text{b}}=0\dots(1)$
Now,
$\vec{\text{c}}.\big(\vec{\text{a}}+\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}+\vec{\text{c}}.\vec{\text{b}}=0+0=0$ [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}+\vec{\text{b}}.$
Again,
$\vec{\text{c}}.\big(\vec{\text{a}}-\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}-\vec{\text{c}}.\vec{\text{b}}=0-0=0$ [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}-\vec{\text{b}}.$
$\Rightarrow\vec{\text{c}}.\vec{\text{a}}=0$ and $\vec{\text{c}}.\vec{\text{b}}=0\dots(1)$
Now,
$\vec{\text{c}}.\big(\vec{\text{a}}+\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}+\vec{\text{c}}.\vec{\text{b}}=0+0=0$ [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}+\vec{\text{b}}.$
Again,
$\vec{\text{c}}.\big(\vec{\text{a}}-\vec{\text{b}}\big)=\vec{\text{c}}.\vec{\text{a}}-\vec{\text{c}}.\vec{\text{b}}=0-0=0$ [From (1)]
So, $\vec{\text{c}}$ is perpendicular to $\vec{\text{a}}-\vec{\text{b}}.$
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
Solve graphically :2x – 3 ≥ 0
For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?
→$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?
Find the area of the triangle with vertices at the points:
$(3, 8), (-4, 2)$ and $(5, -1)$
→$(3, 8), (-4, 2)$ and $(5, -1)$
Find the value of $\cos ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{4}{5}\right)$
→Diffrentiate the following w.r.t.x
→$\log \left(\sqrt{\left.\frac{1+\cos \left(\frac{5 x}{2}\right)}{1-\cos \left(\frac{5 x}{2}\right)}\right)}\right.$
Express the following switching circuit in the symnolic from of logic,. Construct the switching table :
If $\tan ^{-1} x +\tan ^{-1} y +\tan ^{-1} z =\pi$, then show that $\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}=1$
→If $x=\sin t, y=e^{m t}$ then show that $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-m^2 y=0$.
→If a vector$\vec{\text{a}}$ is perpendicular to two non-collinear vectors $\vec{\text{b}}$ and $\vec{\text{c}},$ then show that $\vec{\text{a}}$ is perpendicular to every vector in the plane of $\vec{\text{b}}$ and $\vec{\text{c}}.$
→Evaluate the following:
→$\int x^3 \tan ^{-1} x d x$