Algebra of Vectors

1 Mark Questions

1. Find a vector in the direction of vector <math>2\hat{i} - 3\hat{j} + 6\hat{k}</math> which has magnitude 21 units.

Foreign 2014

? To find a vector in the direction of given vector, first of all we find unit vector in the direction of given vector and then multiply it with given magnitude.

Let <math>\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}</math>

<math>|\vec{a}| = \sqrt{(2)^2 + (-3)^2 + (6)^2}</math> <math>=\sqrt{4+9+36}</math> <math>= \sqrt{49} = 7 \text{ units}</math>

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The unit vector in the direction of the given vector a is

<math display="block">\hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{1}{7} (2\hat{i} - 3\hat{j} + 6\hat{k}) = \frac{2}{7} \hat{i} - \frac{3}{7} \hat{j} + \frac{6}{7} \hat{k}</math>

Therefore, the vector of magnitude equal to 21 units and in the direction of <math>\vec{a}</math> is

<math display="block">21\hat{a} = 21\left(\frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}\right)</math> <math>=6\hat{i}-9\hat{i}+18\hat{k}</math>

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1. Find a vector <math>\overrightarrow{a}</math> of magnitude <math>5\sqrt{2}</math>, making an angle of <math>\frac{\pi}{4}</math> with X-axis, <math>\frac{\pi}{2}</math> with Y-axis and an acute angle <math>\theta</math> with Z-axis. All India 2014

Given, a vector <math>\vec{a}</math> makes an angle <math>\frac{\pi}{4}</math> with

1. axis and <math>\frac{\pi}{2}</math> with Y-axis.

So, <math>l = \cos \frac{\pi}{4}</math> and <math>m = \cos \frac{\pi}{2} \Rightarrow l = \frac{1}{\sqrt{2}}</math>, <math>m = 0</math> We know that, <math>l^2 + m^2 + n^2 = 1</math> <math display="block">\Rightarrow \left(\frac{1}{\sqrt{2}}\right)^2 + (0)^2 + n^2 = 1 \Rightarrow \frac{1}{2} + n^2 = 1</math> <math>\Rightarrow n^2 = 1 - \frac{1}{2} \Rightarrow n = \pm \frac{1}{\sqrt{2}} \Rightarrow n = \frac{1}{\sqrt{2}}</math> <math>\therefore \cos \theta = \frac{1}{\sqrt{2}}</math> [:: <math>\theta</math> is an acute angle with Z-axis] <math>\Rightarrow \theta = \frac{\pi}{4}</math>

Thus, direction cosines of a line are

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<math>\frac{1}{\sqrt{2}}</math>, 0, <math>\frac{1}{\sqrt{2}}</math>

<math>\therefore</math> Vector <math>\vec{a}</math> <math>= |\vec{a}| \left(\cos\frac{\pi}{4}\hat{i} + \cos\frac{\pi}{2}\hat{j} + \cos\frac{\pi}{4}\hat{k}\right)</math> <math display="block">=5\sqrt{2}\left(\frac{1}{\sqrt{2}}\hat{i}+(0)\hat{j}+\frac{1}{\sqrt{2}}\hat{k}\right)=5\hat{i}+5\hat{k}</math> (1/2)

1. Write a unit vector in the direction of the sum of the vectors <math>\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}</math> and

<math>\vec{b} = 2\hat{i} + \hat{i} - 7\hat{k}</math> Delhi 2014C