Question

prove that
P.Q= PxQx + PyQy + PzQz
​

Answer 1

Answer: The answer is dot or scaler product of $\overrightarrow P\,\, and \,\,\overrightarrow Q$ vector

Explanation: To understand the question let us first understand the scaler or dot product of two vectors through the coordinate system.

let us assume that $\overrightarrow P=P_{x} \hat i+P_{y} \hat j+P_{z} \hat k$ and $\overrightarrow Q=Q_{x} \hat i+Q_{y} \hat j+Q_{z} \hat k$

where The components of $\overrightarrow P \,\,and\,\, \overrightarrow Q$ along the x, y and z axis is                      

${P_{x} ,P_{y} \,\,and \,\,P_{z}$  & $Q_{x} ,\,\,Q_{y} \,\,and \,\,Q_{z}$.

Now $\overrightarrow P\,\,.\,\,\overrightarrow Q= (\,P_{x}\hat i+P_{y} \hat j+P_{z} \hat k) . (Q_{x} \hat i+Q_{y} \hat j+Q_{z} \hat k)$

Now after completion of the product there must be nine terms in total , out of which  all the the terms  $\hat i.\hat k\,\,,\hat i.\hat j$ etc. must be equal to zero. As in case of dot product , the scaler product of two unit vector in the perpendicular direction to each other must be zero as following-

$\hat i.\,\hat j=(1) \,(1)\,\,cos\,\,90\textdegree=0$

The remaining three terms containing $\hat i\,.\hat i\,\,,\hat j\,\,.\hat j\,\, and \,\,\hat k.\,\,\hat k$ will give the product as the scaler product of two unit vector in the same direction will be equal to one as follows-

$\hat i.\hat i=(1)\,\,(1)\,\,cos0\textdegree=1$

Therefore,

$\overrightarrow P\,\,.\,\,\overrightarrow Q= (\,P_{x}\hat i+P_{y} \hat j+P_{z} \hat k) . (Q_{x} \hat i+Q_{y} \hat j+Q_{z}\hat k)\\\\\overrightarrow P\,\,.\,\,\overrightarrow Q=\,P_{x}Q_{x}+P_{y}Q_{y}+P_{z}Q_{z}$   hence proved

Answer 2

Answer:

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