Let \vec{a}= \hat{i}+2\hat{j}-3\hat{k} and \vec{b}= 3\hat{i}-\hat{j}+2\hat{k}  be two vectors. Show that the vectors \left ( \vec{a}+\vec{b} \right ) and \left ( \vec{a}-\vec{b} \right ) are perpendicular to each other.

 

 

 

 
 
 
 
 

Answers (1)

Now \left ( \vec{a}+\vec{b} \right )\cdot \left ( \vec{a}-\vec{b} \right )= \vec{a}\cdot \vec{a}-\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{a}-\vec{b}\cdot \vec{b}
                                             = \left | a \right |^{2}-\left | \vec{b} \right |^{2}={\left | \hat{i}+2\hat{j}-3\hat{k} \right |}^2-{\left | 3\hat{i}-\hat{j}+2\hat{k} \right |}^2
                                             = \left ( 1+4+9 \right )-\left ( 9+1+4 \right )= 0
Hence, \vec{a}+\vec{b}  & \vec{a}-\vec{b}  are perpendicular to each other. 

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