Product of Two Vectors > Vector (Cross) Product

If \[\vec{a}\] and \[\vec{b}\] are two non-zero vectors vectors with angle \[\theta\] between them, then their vector product is: \[\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \hat{n}\]

where \[\hat{n}\] is a unit vector perpendicular to both \[\vec{a}\] and \[\vec{b}\], in the direction given by the right-hand rule.

Cross Product Angle: \[\sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|}\]

• Not commutative: \[\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})\].

• Distributive over addition.

• \[\vec{a} \times \vec{a} = \vec{0}\].

• The result is perpendicular to both vectors.

• If vectors are parallel, then \[\sin \theta = 0\], so \[\vec{a} \times \vec{b} = \vec{0}\].

• If vectors are perpendicular, then \[|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\].

• Unit Vector Cross Product Relations: 
  \[\hat{i} \times \hat{j} = \hat{k}\]
  \[\hat{j} \times \hat{k} = \hat{i}\]
  \[\hat{k} \times \hat{i} = \hat{j}\]

  and

  \[\hat{j} \times \hat{i} = -\hat{k}\]
  \[\hat{k} \times \hat{j} = -\hat{i}\]
  \[\hat{i} \times \hat{k} = -\hat{j}\]

• Determinant form: 

  \[\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\]

Find \[|\vec{a} \times \vec{b}|\], if \[\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\] and \[\vec{b} = 3\hat{i} + 5\hat{j} - 2\hat{k}\]

Solution: We have

Hence \[|\vec{a} \times \vec{b}| = \sqrt{(-17)^2 + (13)^2 + (7)^2} = \sqrt{507}\]

Find the area of a parallelogram whose adjacent sides are given by the vectors \[\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}\] and \[\vec{b} = \hat{i} - \hat{j} + \hat{k}\].

Solution: The area of a parallelogram with \[\vec{a}\] and \[\vec{b}\] as its adjacent sides is given by \[|\vec{a} \times \vec{b}|\].

Now

Therefore \[|\vec{a} \times \vec{b}| = \sqrt{25 + 1 + 16} = \sqrt{42}\]

and hence, the required area is \[\sqrt{42}\].

• The cross product of two vectors is a vector quantity.
• The cross product uses the sine of the angle between the vectors.
  \[ \vec{a} \times \vec{b} = |\vec{a}||\vec{b}| \sin\theta\,\hat{n} \]
• The resulting vector is perpendicular to both vectors.
• The cross product is useful in area and direction problems. 
• For two non-zero vectors, \[ \vec{a} \times \vec{b} = \vec{0} \] indicates that the vectors are parallel or collinear.
• Area of Triangle: \[ \frac{1}{2}|\vec{a} \times \vec{b}| \]
•  Area of Parallelogram: \[ |\vec{a} \times \vec{b}| \]