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### Section 11.2 : Vector Arithmetic

5. Determine if $$\vec a = \left\langle {3, - 5,1} \right\rangle$$ and $$\vec b = \left\langle {6, - 2,2} \right\rangle$$ 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,

$\vec a = k\,\vec b$ Show Step 2

Let’s just take a look at the first component from each vector. It is obvious that $$6 = 2\left( 3 \right)$$. So, to convert the first components we’d need to multiply $$\vec a$$ by 2. However, if we did that we’d get,

$2\vec a = \left\langle {6, - 10,2} \right\rangle \ne \vec b$

This is clearly not $$\vec b$$. The first component is correct and the third component is correct but the second isn’t correct. Therefore, there is no single number, $$k$$, that we can use to convert $$\vec a$$ into $$\vec b$$ through scalar multiplication.

This in turn means that $$\vec a$$ and $$\vec b$$ cannot possibly be parallel.