Question

Determine the sum of two vectors.

Short answer

The sum of vectors is \(\langle 2,4\rangle \).

The geometrical illustration of sum of vectors as shown in Figure:

Step-by-step solution

Given data

Vectors \(\langle 3, - 1\rangle \) and \(\langle - 1,5\rangle \).

Concept of vectors

Vector is a quantity that has both magnitude and direction. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity's magnitude.

Describe the process and draw the Vectors

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

Write the expression for vector some of vectors.

\(a + b = \left\langle {{a_1} + {b_1},{a_2} + {b_2}} \right\rangle \)

Substitute 3 for \({a_1}, - 1\) for \({b_1}, - 1\) for \({a_2}\), and 5 for \({b_2}\) in equation (1),

\(\begin{aligned}{l}a + b = \langle 3 - 1, - 1 + 5\rangle \\a + b = \langle 2,4\rangle \end{aligned}\)

Locate the first point \((3, - 1)\) in xy-plane and connect a line from origin to point \((3, - 1)\) to plot vector \(\langle 3, - 1\rangle \). Locate the second point \(( - 1,5)\) on xy-plane and connect a line at terminal point of vector \(\langle 3, - 1\rangle \) and to point \(( - 1,5)\).

This is second vector \(\langle - 1,5\rangle \).

Sketch a vector by connecting initial point of vector \(\langle 3, - 1\rangle \) and terminal point of vector \(\langle - 1,5\rangle \).

This line is the resultant sum of two vectors in accordance to triangle law.

From explanation, the geometrical illustration of sum of vectors as shown in Figure: