Question

Let vector a=i-j,b=3j-k and c=7i-k.Find a vector d which is perpendicular to both a and b and c.d=1

Answer 1

Here we use the fact that scalar product of perpendicular vectors is zero. Rest is just algebra.

Answer 2

Therefore $\overrightarrow d=\frac{1}{4} \hat{i}+\frac{1}{4} \hat{j}+\hat{k}$

Step-by-step explanation:

Given;

$\overrightarrow a=\hat{i}-\hat{j}$ ,  $\overrightarrow b=3\hat{j}-\hat{k}$ and vector $\overrightarrow c=7\hat{i}-\hat{k}$

Find a vector $d$ which is perpendicular to both $a$ and $b$ and $c.d=1$

Let,

$\overrightarrow d=x\hat{i}+y\hat{j}+z\hat{k}$

∵ vector $d$ which is perpendicular to both $a$ and $b$

Then;

  $d.a=0$

⇒$x-y=0$

⇒$x=y$ __equation-1

And,

  $d.b=0$

⇒$3y-z=0$

⇒$z=3y$__equation-2

Also given,

  $c.d=1$

⇒$7x-z=1$

From equation-2,

⇒$7x-3y=1$

From equation-1,

⇒$7x-3x=1$

⇒$4x=1$

∴$x=y=\frac{1}{4}$

And $z=4\times \frac{1}{4} =1$

So $\overrightarrow d=\frac{1}{4} \hat{i}+\frac{1}{4} \hat{j}+\hat{k}$