Show that the vectors v1 = 4i − 6j + 9k and v2 = −6i + 9j − 27/2k are parallel.

Question

ScanSolve AI Tutor · Accepted Answer

To show that two vectors are parallel, we need to demonstrate that one vector is a scalar multiple of the other. That is, if $\vec{a}$ and $\vec{b}$ are two vectors, they are parallel if $\vec{a} = c\vec{b}$ for some scalar $c$ .

Define the given vectors.
Let $\vec{v_1} = 4\vec{i} - 6\vec{j} + 9\vec{k}$
Let $\vec{v_2} = -6\vec{i} + 9\vec{j} - \frac{27}{2}\vec{k}$
Assume $\vec{v_2} = c\vec{v_1}$ and find the scalar $c$ .
We set the components equal to each other:
$-6\vec{i} + 9\vec{j} - \frac{27}{2}\vec{k} = c(4\vec{i} - 6\vec{j} + 9\vec{k})$ $-6\vec{i} + 9\vec{j} - \frac{27}{2}\vec{k} = 4c\vec{i} - 6c\vec{j} + 9c\vec{k}$
Equate the coefficients for each component.
For the $\vec{i}$ component:
$-6 = 4c \implies c = \frac{-6}{4} = -\frac{3}{2}$
For the $\vec{j}$ component:
$9 = -6c \implies c = \frac{9}{-6} = -\frac{3}{2}$
For the $\vec{k}$ component:
$-\frac{27}{2} = 9c \implies c = \frac{-27/2}{9} = \frac{-27}{2 \times 9} = \frac{-27}{18} = -\frac{3}{2}$
Conclude based on the scalar value.
Since the scalar $c = -\frac{3}{2}$ is consistent for all components, the vectors are parallel.
Therefore, $\vec{v_2} = -\frac{3}{2}\vec{v_1}$ .

The vectors $4\vec{i} - 6\vec{j} + 9\vec{k}$ and $-6\vec{i} + 9\vec{j} - \frac{27}{2}\vec{k}$ are indeed parallel.
$\boxed{\text{The vectors are parallel because } \vec{v_2} = -\frac{3}{2}\vec{v_1}}$

Show that the vectors v1 = 4i − 6j + 9k and v2 = -6i + 9j - (27)/(2)k are parallel.

Step-by-step solution

Define the given vectors.

Assume $\vec{v_2} = c\vec{v_1}$ and find the scalar $c$ .

Equate the coefficients for each component.

Conclude based on the scalar value.

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