Question

Vector u has its initial point at (21, 12) and its terminal point at (19, -8). Vector v has a direction opposite that of u, whose magnitude is five times the magnitude of v. Which is the correct form of vector v expressed as a linear combination of the unit vectors i and j?

Answer 1

$\boxed{\vec{v}=\frac{2}{5}i+4j}$

Explanation:

In this exercise, we have the following facts for the vector $\vec{u}$:

• It has its initial point at $(21,12)$, let's call it $P_{1}$
• It has its terminal point at $(19,-8)$, let's call it $P_{2}$

Since the vector $\vec{u}$ goes from point $P_{1}$ to $P_{2}$, then:

$\vec{u}=(19,-8)-(21,12) \\ \\ \vec{u}=(19-21,-8-12) \\ \\ \vec{u}=(-2,-20)$

On the other hand, we have the following facts for the vector $\vec{v}$:

• Vector $\vec{v}$ has a direction opposite that of $\vec{u}$,
• The magnitude of $\vec{u}$ is five times the magnitude of $v$.

So we can write this relationship as follows:

$5\vec{v}=-\vec{u} \\ \\ \vec{v}=-\frac{1}{5}\vec{u} \\ \\ \vec{v}=-\frac{1}{5}(-2,-20) \\ \\ \vec{v}=(\frac{2}{5},4) \\ \\ \\ Finally: \\ \\ \boxed{\vec{v}=\frac{2}{5}i+4j}$

Learn more:

Length of vectors: https://brainly.com/question/12264340

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