Multiplication Of Vectors And Scalar Quantity - Practice Questions & MCQ

If a is a vector and "λ" is a scalar (i.e. a real number), then λa is a vector whose magnitude is |λ| times that of a and whose direction is the same (or opposite) that of a according as the value of λ is positive (or negative).

Note that λa is collinear to the vector a. 

If

$
\tilde{\mathbf{a}}=\mathrm{a}_1 \hat{\mathbf{i}}+\mathrm{a}_2 \hat{\mathbf{j}}+\mathrm{a}_3 \hat{\mathbf{k}}
$
then,
$
\lambda \tilde{\mathbf{a}}=\left(\lambda \mathrm{a}_1\right) \hat{\mathbf{i}}+\left(\lambda \mathrm{a}_2\right) \hat{\mathbf{j}}+\left(\lambda \mathrm{a}_3\right) \hat{\mathbf{k}}
$

Properties of multiplication of a vector by a scalar

a and b are vectors, λ  and γ are scalars.

1. $\quad \lambda(-\tilde{\mathbf{a}})=(-\lambda)(\tilde{\mathbf{a}})=-(\lambda \tilde{\mathbf{a}})$
2. $\quad(-\lambda)(-\tilde{\mathbf{a}})=\lambda \tilde{\mathbf{a}}$
3. $\quad \lambda(\gamma \tilde{\mathbf{a}})=(\lambda \gamma)(\tilde{\mathbf{a}})=\gamma(\lambda \tilde{\mathbf{a}})$
4. $\quad(\lambda+\gamma) \tilde{\mathbf{a}}=\lambda \tilde{\mathbf{a}}+\gamma \tilde{\mathbf{a}}$
5. $\quad \lambda(\tilde{\mathbf{a}}+\tilde{\mathbf{b}})=\lambda(\tilde{\mathbf{a}})+\lambda(\tilde{\mathbf{b}})$

A geometric visualization of the multiplication of a vector by a scalar