Question

Consider the two vectors $\overrightarrow{\mathrm{A}}=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}$and$\overrightarrow{\mathrm{B}}=-\hat{\mathrm{i}}-4\hat{\mathrm{j}}$ calculate (a)$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ (b) $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$(c) $\left|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\right|$(d) $\left|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}\right|$and (e) the directions of $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$and $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$.

Short answer

$$\begin{gathered} \mathrm{a})\left(2\hat{\mathrm{i}}-6\hat{\mathrm{j}}\right) \\ \mathrm{b})\left(4\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right) \\ \mathrm{c})6.32 \\ \mathrm{d})4.47 \\ \mathrm{e})288°\mathrm{and}26.6° \end{gathered}$$

Step-by-step solution

Define Vector Components

In a two-dimensional coordinate system, the x-component andy-component are commonly considered to be the components of a vector. It can be written as $V=\left(vx,vy\right)$with V denoting the vector.

These are the components of vectors created along the axes. In this article, we shall find the components of any given vector using formulas for both two-dimensional and three-dimensional coordinate systems.

$$\begin{gathered} v_x=V\cos \theta \\ {v}_y=V\sin \theta \end{gathered}$$

State the given data

$$\begin{gathered} \overrightarrow{A}=3\hat{\text{i}}\text{- 2}\hat{\text{j}} \\ \overrightarrow{B}=-1\hat{\text{i}}\text{- 4}\hat{\text{j}} \end{gathered}$$

(a) Find

$$\begin{gathered} \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\left(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\right)+\left(-1\hat{\mathrm{i}}-4\hat{\mathrm{j}}\right) \\ =\left(2\hat{\mathrm{i}}-6\hat{\mathrm{j}}\right) \end{gathered}$$

Hence, $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\left(2\hat{\mathrm{i}}-6\hat{\mathrm{j}}\right)$

(b) Find

$$\begin{gathered} \overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}=\left(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\right)-\left(-1\hat{\mathrm{i}}-4\hat{\mathrm{j}}\right) \\ =\left(4\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right) \end{gathered}$$

Hence,role="math" localid="1663597953547" $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}=\left(4\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right)$

(c) Find

$$\begin{gathered} \left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{2^2+{\left(-6\right)}^2} \\ =6.32 \end{gathered}$$

Hence,$\left|\overrightarrow{A}+\overrightarrow{B}\right|=6.32$

(d) Find

$$\begin{gathered} \left|\overrightarrow{A}-\overrightarrow{B}\right|=\sqrt{4^2+{\left(2\right)}^2} \\ =4.47 \end{gathered}$$