Vector Addition and Subtraction - Practice Questions & MCQ

Since each vector may have its own direction, the process of addition of vectors is different from adding two scalars. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in fig (a). 

Suppose, for example, that both vectors represent displacement. If an object moves first from the initial point to the terminal point of vector , then from the initial point to the terminal point of vector , the overall displacement is the same as if the object had made just one movement from the initial point of  to the terminal point of the vector . Thus  joins starting point of one vector to terminal point of other vector when they are placed one after the other. For obvious reasons, this approach is called the triangle method

Parallelogram Law of Addition

A second method for adding vectors is called the parallelogram method. With this method, we place the two vectors so they have the same initial point, and then we draw a parallelogram with the vectors as two adjacent sides, as in fig (b)

Here the sum of the vectors is given by the vector along the diagonal that passes through the common starting point of both the vectors.

Polygon law of addition

If a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

$$
\vec{a}+\vec{b}+\vec{c}+\vec{d}+\vec{e}=-\vec{f}
$$

OR,
$$
\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{CD}}+\overrightarrow{\mathrm{DE}}+\overrightarrow{\mathrm{EF}}=\overrightarrow{\mathrm{AF}}
$$

Subtraction of Vectors

If a and b are two vectors, then their subtraction or difference  is defined as $\overrightarrow{\mathbf{a}}+(-\overrightarrow{\mathbf{b}})$, where  is the negative of vector b having equal magnitude but opposite direction that of b. Graphically, it is depicted by drawing a vector from the terminal point of b to the terminal point of a.

If the vectors are defined in terms of $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$, i.e.,
$$
\vec{a}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}} \text { and } \vec{b}=b_1 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}+b_3 \hat{\mathbf{k}}
$$
then their sum is defined as,
$$
\vec{a}+\vec{b}=\left(a_1+b_1\right) \hat{\mathbf{i}}+\left(a_2+b_2\right) \hat{\mathbf{j}}+\left(a_3+b_3\right) \hat{\mathbf{k}}
$$

If,
$$
\vec{a}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}} \text { and } \vec{b}=b_1 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}+b_3 \hat{\mathbf{k}}
$$
then their difference is defined as,
$$
\vec{a}-\vec{b}=\left(a_1-b_1\right) \hat{\mathbf{i}}+\left(a_2-b_2\right) \hat{\mathbf{j}}+\left(a_3-b_3\right) \hat{\mathbf{k}}
$$

Properties of vector addition
The sum of two vectors is always a vector.
1. $\vec{a}+\vec{b}=\vec{b}+\vec{a}$
(Commutative property)
2. $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
(Associative property)
3. $\vec{a}+\overrightarrow{0}=\overrightarrow{0}+\vec{a}=\vec{a}$ (additive identity)
4. $\vec{a}+(-\vec{a})=(-\vec{a})+\vec{a}=\overrightarrow{0}$ (additive inverse)

Properties of vector Subtraction
1. $\vec{a}-\vec{b} \neq \vec{b}-\vec{a}$
2. $(\vec{a}-\vec{b})-\vec{c} \neq \vec{a}-(\vec{b}-\vec{c})$
3. For any two vectors $\vec{a}$ and $\vec{b}$
(a) $|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|$
(b) $|\vec{a}+\vec{b}| \geq||\vec{a}|-|\vec{b}||$
(c) $|\vec{a}-\vec{b}| \leq|\vec{a}|+|\vec{b}|$
(d) $|\vec{a}-\vec{b}| \geq||\vec{a}|-|\vec{b}||$