Question

To determine the sum of two three-dimensional vectors.

Short answer

The value of the sum of vectors is \(\underline {\langle 3,8,1\rangle } \) and is illustrated geometrically.

Step-by-step solution

Given data

Vectors are \(\langle 3,0,1\rangle \) and \(\langle 0,8,0\rangle \).

Definition of Sum of Vectors and Triangle Law

Sum of vectors:

Consider the two three-dimensional vectors\(a = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \)and\({\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \).

The vector sum of two vectors is,\(a + b = \left\langle {{a_1} + {b_1},{a_2} + {b_2},{a_3} + {b_3}} \right\rangle \). ...… (1)

Triangle law:

Consider two vectors\(u\)and then vector\(v\).

Draw the vector\(u\)first and then the second vector\(u\)at the end of vector\(u\), then the resultant vector is sum of two vectors\(({\bf{u}} + {\bf{v}})\).

Substitute the values in the equation

Substitute 3 for \({a_1},0\), \({b_1},{\rm{ }}0 \to {a_1},\;0 \to {b_1},\;0 \to {a_2},\;8 \to {b_2},\;1 \to {a_3}\), and \(0 \to {b_3}\) in equation (1).

\(\begin{aligned}{c}a + b = \langle 3 + 0,0 + 8,1 + 0\rangle \\ = \langle 3,8,1\rangle \end{aligned}\)

Locate the first point\((3,0,1)\)in\(xyz\)-plane and connect a line from origin to point\((3,0,1)\)to plot vector\(\langle 3,0,1\rangle \).

Locate the second point\((0,8,0)\)on the\(xy\)-plane and connect a line at the terminal point of vector\(\langle 3,0,1\rangle \)to the point\((0,8,0)\).

This is the second vector\(\langle 0,8,0\rangle \).

Sketch a vector by connecting the initial point of vector\(\langle 3,0,1\rangle \)and terminal point of vector\(\langle 0,8,0\rangle \).

By triangle law, this line is the resultant sum of the two vectors.

From explanation, draw the geometrical illustration of sum of vectors as shown in Figure 1.

Thus, the value of the sum of vectors is \(\underline {\langle 3,8,1\rangle } \) and is illustrated geometrically.