Question

To determine whether the given vectors are orthogonal, parallel, or neither.

(a)For vector\(u = \langle - 3,9,6\rangle \)and\(v = \langle 4, - 12, - 8\rangle \)

(b)For vector\(u = \langle 1, - 1,2\rangle \)and\(v = \langle 2, - 1,1\rangle \)

(c)For vector\({\rm{u}} = \langle a,b,c\rangle \)and\({\rm{v}} = \langle - b,a,0\rangle \)

Short answer

(a) The vectors \(u\) and \(v\) are parallel.

(b) The vectors \(u\) and \(v\) are neither parallel nor orthogonal.

(c) The vectors \({\rm{u}}\) and \({\rm{v}}\) are orthogonal.

Step-by-step solution

Theorem used for Dot Product

Definition used:

"If two vectors\({\rm{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \)and\({\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \), then their dot product is\({\rm{a}} \cdot {\rm{b}} = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\)".

Theorem used:

1. Two vectors\({\rm{a}} = \leStep 2: Calculation to check whether the vectors are orthogonal, parallel or neitherzft\langle {{a_1},{a_2},{a_3}} \right\rangle \)and\({\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \)are orthogonal if and only if\({\bf{a}} \cdot {\bf{b}} = 0\).

2. Two vectors \({\rm{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \) and \({\rm{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \) are parallel, if and only if \(\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}} = \frac{{{a_3}}}{{{b_3}}}\).

Calculation to check whether the vectors are orthogonal, parallel or neither

The given two vectors are \(u = \langle - 3,9,6\rangle \) and \(v = \langle 4, - 12, - 8\rangle \).

In order to check whether the above two vectors are parallel, orthogonal, or neither, use the theorems mentioned above.

Calculate the ratio of the components of two vectors as follows.

\(\begin{aligned}{l}\frac{{ - 3}}{4} &= \frac{9}{{ - 12}} &= \frac{6}{8}\\\frac{{ - 3}}{4} &= \frac{{ - 3}}{4} &= \frac{{ - 3}}{4}\end{aligned}\)

Thus, by theorem\(\left( 2 \right)\), vectors\(u\)and \(v\) are parallel.

Calculation to check whether the vectors are orthogonal, parallel or neither

The given two vectors are \(u = \langle 1, - 1,2\rangle \) and \(v = \langle 2, - 1,1\rangle \).

In order to check whether the above two vectors are parallel, orthogonal, or neither, use the theorems mentioned above.

Apply the definition with \({a_1} = 1,{a_2} = - 1,{a_3} = 2,{b_1} = 2,{b_2} = - 1\), and \({b_3} = 1\) and obtain the dot product of two vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= (1)(2) + ( - 1)( - 1) + (2)(1)\\{\rm{a}} \cdot {\rm{b}} &= 2 + 1 + 2\\{\rm{a}} \cdot {\rm{b}} &= 5\\{\rm{a}} \cdot {\rm{b}} \ne 0\end{aligned}\)

Thus, by theorem\(\left( 1 \right)\), vectors\(u\)and \(v\) are orthogonal.

Calculate the ratio of the components of two vectors as \(\frac{1}{2} = \frac{{ - 1}}{{ - 1}} = \frac{2}{1}\).

Thus, by theorem\(\left( 2 \right)\), vectors\(u\)and \(v\) are not parallel.

Thus, the vectors \(u\) and \(v\) are neither parallel nor orthogonal.

Calculation to check whether the vectors are orthogonal, parallel or neither

The given two vectors are \({\rm{u}} = \langle a,b,c\rangle \) and \({\rm{v}} = \langle - b,a,0\rangle \).

In order to check whether the above two vectors are parallel, orthogonal, or neither, use the theorems mentioned above.

Apply the formula with \({a_1} = a,{a_2} = b,{a_3} = c,{b_1} = - b,{b_2} = a\), and \({b_3} = 0\) and obtain the dot product of two vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= (a)( - b) + (b)(a) + (c)(0)\\{\rm{a}} \cdot {\rm{b}} &= - ab + ba + 0\\{\rm{a}} \cdot {\rm{b}} &= - ab + ab\\{\rm{a}} \cdot {\rm{b}} &= 0\end{aligned}\)

Thus, by theorem\(\left( 1 \right)\), vectors\(u\)and \(v\) are orthogonal.