Multiplication in Vector Algebra

Multiplication of a vector by a scalar is one of the most basic ideas in Vector Algebra. It explains how a vector changes when its magnitude is enlarged, reduced, or reversed, while its line of action remains the same.

If \[\vec{a}\] is a vector and \[\lambda\] is a scalar, then \[\lambda\vec{a}\] is called the multiplication of the vector \[\vec{a}\] by the scalar \[\lambda\]. The resulting quantity is also a vector, and it is collinear with \[\vec{a}\].

• \[\lambda\vec{a}\] is always a vector collinear with \[\vec{a}\].

• If \[\lambda > 0\], the direction remains the same.

• If \[\lambda < 0\], the direction becomes opposite.

• If \[\lambda = 0\], the result is the zero vector.

• Magnitude: \[|\lambda\vec{a}| = |\lambda| |\vec{a}|\]

• Negative of a Vector: When \[\lambda = -1\],
  \[-\vec{a} = (-1)\vec{a}\]

• Unit Vector: For a non-zero vector \[\vec{a}\],

  \[\hat{a} = \frac{\vec{a}}{|\vec{a}|}\]

Question:

Let \[\vec{b} = -\hat{i} + 4\hat{j}\]. Find \[-\frac{1}{2}\vec{b}\]. Comment on the direction.

Solution:

• Magnitude is multiplied by \[|-\frac{1}{2}| = \frac{1}{2}\], so it becomes half the length.

• Since the scalar is negative, the direction of \[-\frac{1}{2}\vec{b} \]is opposite to that of \[\vec{b}\].

• Multiplication of a vector by a scalar gives a new vector collinear with the original.

• Magnitude scales by \[|\lambda|\]; direction depends on the sign of \[\lambda\].

• \[\lambda > 0\]: Same direction; \[\lambda < 0\]: Opposite direction.

• \[\lambda = 0\]: Result is the null vector.

• Used extensively in components form: multiply each component by the scalar.