Calculus II - Vector Arithmetic

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Section 11.2 : Vector Arithmetic

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6. Determine if \(\vec v = 9\vec i - 6\vec j - 24\vec k\) and \(\vec w = \left\langle { - 15,10,40} \right\rangle \) are parallel vectors.

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Recall that two vectors are parallel if they are scalar multiples of each other. In other words, these two vectors will be scalar multiples if we can find a number \(k\) such that,

\[\vec v = k\,\vec w\] Show Step 2

Let’s just take a look at the first component from each vector. It is should be clear that \( - 15 = \left( { - \frac{5}{3}} \right)\left( 9 \right)\). So, to convert the first components we’d need to multiply \(\vec v\) by \( - \frac{5}{3}\) .

if we did that we’d get,

\[ - \frac{5}{3}\vec v = \left\langle { - 15,10,40} \right\rangle = \vec w\]

So, we were able to find a number \(k\) that we could use to convert \(\vec v\) into \(\vec w\) through scalar multiplication and so the two vectors are parallel.