Product of Two Vectors > Scalar (Dot) Product

Vectors can be combined in more than one way, and each type of product gives different information. The two most important products are the scalar (dot) product and the vector (cross) product, which are used to find angles, projections, area, and direction-related results.

If \[\vec{a}\] and \[\vec{b}\] are two vectors and \[\theta\] is the angle between them, then their scalar product is given by:

• Commutative: \[\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\]

• Distributive: \[\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}\]

• \[\vec{a} \cdot \vec{a} = |\vec{a}|^2\]

• If \[\vec{a} \cdot \vec{b} = 0\], the vectors are perpendicular if both are non-zero.

• \[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]

• \[\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1\]

  \[\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0\]

Projection is the part of one vector in the direction of another vector.

Scalar projection of \[\vec{a}\] on \[\vec{b}\]

Vector projection of \[\vec{a}\] on \[\vec{b}\]

For any two vectors \[\vec{a}\] and \[\vec{b}\], we always have \[|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|\] (triangle inequality).

                Fig 10.21

Solution: The inequality holds trivially in case either \[\vec{a} = \vec{0}\] or \[\vec{b} = \vec{0}\] (How?). So, let \[|\vec{a}| \neq 0 \neq |\vec{b}|\]. Then,

\[|\vec{a} + \vec{b}|^2 = (\vec{a} + \vec{b})^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})\]

\[= \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b}\]

\[= |\vec{a}|^2 + 2\vec{a} \cdot \vec{b} + |\vec{b}|^2\] (scalar product is commutative)

\[\leq |\vec{a}|^2 + 2|\vec{a} \cdot \vec{b}| + |\vec{b}|^2\] (since \[x \leq |x| \forall x \in \mathbf{R}\])

\[\leq |\vec{a}|^2 + 2|\vec{a}| |\vec{b}| + |\vec{b}|^2\] (from Example 19)

\[= (|\vec{a}| + |\vec{b}|)^2\]

Hence \[|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|\]

• The dot product of two vectors is a scalar quantity.
• The dot product uses the cosine of the angle between the vectors.
  \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos\theta \]
• The dot product is useful in finding the angle between two vectors and projection.
• For two non-zero vectors, \[ \vec{a} \cdot \vec{b} = 0 \] indicates that the vectors are perpendicular.