Question

To find the sum of two vectors.

Vectors \(\left( {1,3, - 2} \right)\) and \(\left( {0,0,6} \right)\)

Short answer

The sum of vectors is \(\left( {1,3,4} \right)\).

Step-by-step solution

Concept of Sum of Vectors

Sum of vectors:

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

The vector sum of two vectors is,

\(a + b = \left\langle {{a_1} + {b_1},{a_2} + {b_2},{a_3} + {b_3}} \right\rangle \)

 Step 2: Calculation of the sum of the two vectors

Triangle law:

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

Substitute for \({a_1},0\) for \({b_1},3\) for \({a_2},0\) for \({b_2}, - 2\) for \({a_3}\), and\(6\)for \({b_3}\) in equation of vector sum of two vectors,

\(\begin{aligned}{l}a + b = \langle 1 + 0,3 + 0, - 2 + 6\rangle \\a + b = \langle 1,3,4\rangle \end{aligned}\)

Locate the first point \((1,3, - 2)\) in \(x{\rm{ }}y{\rm{ }}z\)-plane and connect a line from origin to point\(\langle 1,3, - 2\rangle \)to plot vector \((1,3, - 2)\). Locate the second point \((0,0,6)\) on \(x{\rm{ }}y\)-plane and connect a line at terminal point of vector\((1,3, - 2)\)and to point \((0,0,6)\). This is second vector \((0,0,6)\).

Draw the geometric illustration of sum of vectors

Sketch a vector by connecting initial point of vector \((1,3, - 2)\) and terminal point of vector \((0,0,6)\). This line is the resultant sum of two vectors according to triangle law.

Thus, the value of sum of vectors is \((1,3, - 2)\)and illustrated geometrically.