Question

If A = 3i + 2j and B = 2i + 3j - k , then find a unit vector along ( A-B ) . ( all these are vectors

Answer 1

Answer:

$\dfrac{1\hat i-1\hat j+1\hat k}{\sqrt{3}}.$

Explanation:

Given vectors are:

$\vec A = 3\hat i+2\hat j.\\\vec B = 2\hat i+3\hat j-\hat k.$

where,

$\hat i,\ \hat j,\ \hat k$ are the unit vectors along x, y and z axes respectively.

The vector $\vec A-\vec B$ is given by

$\vec A-\vec B =(3\hat i+2\hat j)-(2\hat i+3\hat j-\hat k)\\=(3-2)\hat i+(2-3)\hat j-(-1)\hat k\\=1\hat i-1\hat j+1\hat k.$

The magnitude of this vector is given by

$|\vec A-\vec B|=\sqrt{(1)^2+(-1)^2+(1)^2}=\sqrt 3.$

The unit vector is defined as that vector which represents the direction only and it has unit magnitude.

The unit vector along some vector $\vec P$ is given by

$\hat P = \dfrac{\vec P}{|\vec P|}.$

Thus, the unit vector along $\vec A-\vec B$ is given by

$\hat n = \dfrac{\vec A-\vec B}{|\vec A-\vec B|}=\dfrac{1\hat i-1\hat j+1\hat k}{\sqrt{3}}.$

Answer 2

find the value of b the vectors are collinear 6i vector+ 2jvector +12 k vector and ivector + bj vector +2k vector