What is Dot Product of Two Vectors ?

Let \(\vec{a}\) and \(\vec{b}\) be two non-zero vectors inclined at an angle \(\theta\). Then the scalar product or dot product of two vectors, \(\vec{a}\) with \(\vec{b}\) is denoted by \(\vec{a}\).\(\vec{b}\) and is defined as,

\(\vec{a}\).\(\vec{b}\) = \(|\vec{a}||\vec{b}|cos\theta\) 

If \(\vec{a}\) = \(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\vec{b}\) = \(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\). Then

\(\vec{a}\).\(\vec{b}\) = \(a_1b_1+a_2b_2+c_1c_2\)

Properties of Dot Product of Two Vectors :

(a) \(\vec{a}\).\(\vec{b}\) = \(|\vec{a}||\vec{b}|cos\theta\) (0 \(\le\) \(\theta\) \(\le\) \(\pi\))

Note that if \(\theta\) is acute then \(\vec{a}\).\(\vec{b}\) > 0  if \(\theta\) is obtuse then \(\vec{a}\).\(\vec{b}\) < 0

(b) (i)  \(\vec{a}\).\(\vec{a}\) = \({|\vec{a}|}^2\)  (ii)  \(\vec{a}\).\(\vec{b}\) = \(\vec{b}\).\(\vec{a}\)  (Commutative)

(c)  \(\vec{a}\).(\(\vec{b}\) + \(\vec{c}\)) = \(\vec{a}\).\(\vec{b}\) + \(\vec{a}\).\(\vec{c}\)  (Distributive)

(d) \(\vec{a}\).\(\vec{b}\) = 0  \(\iff\)  \(\vec{a}\)\(\perp\)\(\vec{b}\)  (\(\vec{a}\), \(\vec{b}\)) \(\ne\) 0

(e)  \(\vec{i}\).\(\vec{i}\) = \(\vec{j}\).\(\vec{j}\) = \(\vec{k}\).\(\vec{k}\) = 1  \(\vec{i}\).\(\vec{j}\) = \(\vec{j}\).\(\vec{k}\) = \(\vec{k}\).\(\vec{i}\) = 0

(f)  Projection of \(\vec{a}\) on \(\vec{b}\) = \(\vec{a}.\vec{b}\over |\vec{b}|\).  Provided \(|\vec{b}|\) \(\ne\) 0

Note :

(i) The vector component of \(\vec{a}\) along \(\vec{b}\) = (\(\vec{a}.\vec{b}\over {\vec{b}}^2\))\(\vec{b}\) and perpendcular to \(\vec{b}\) = \(\vec{a}\) – (\(\vec{a}.\vec{b}\over {\vec{b}}^2\))\(\vec{b}\).

(ii) Maximum Value of \(\vec{a}\).\(\vec{b}\) = |\(\vec{a}\)||\(\vec{b}\)|

(iii) Minimum Value of \(\vec{a}\).\(\vec{b}\) = -|\(\vec{a}\)||\(\vec{b}\)|

Example : Find dot product of vectors \(\vec{a}\) = \(2\hat{i}+2\hat{j}-\hat{k}\) and \(\vec{b}\) = \(6\hat{i}-3\hat{j}+2\hat{k}\)

Solution : We have \(\vec{a}\) = \(2\hat{i}+2\hat{j}-\hat{k}\) and \(\vec{b}\) = \(6\hat{i}-3\hat{j}+2\hat{k}\)

\(\vec{a}\).\(\vec{b}\) = (\(2\hat{i}+2\hat{j}-\hat{k}\)).(\(6\hat{i}-3\hat{j}+2\hat{k}\))

= (2)(6) + (2)(-3) + (-1)(2) = 12 – 6 – 2 = 4

Hope you learnt what is dot product of two vectors formula and it properties, learn more concepts of vectors and practice more questions to get ahead in the competition. Good luck!

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