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Section 5-3 : Dot Product

7. Determine if \(\vec a = \left\langle {3,10} \right\rangle \) and \(\vec b = \left\langle {4, - 1} \right\rangle \) are parallel, orthogonal or neither.

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Based on a quick inspection of the components we can see that the first components of the vectors have the same sign and the second have opposite signs. This means there is no possible way for these two vectors to be scalar multiples since there is no number that will change the sign on the second components and leave the sign of the first component unchanged.

Therefore, we can quickly see that the two vectors are not parallel.

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Let’s do a quick dot product on the two vectors next.

\[\vec a\centerdot \vec b = 2\] Okay, the dot product is not zero and we know from the notes that this in turn means that the two vectors are not orthogonal.

The answer to the problem is therefore the two vectors are neither parallel or orthogonal.

On a side note an alternate method for working this problem is to find the angle between the two vectors and using that to determine the answer.

Depending on which method you find easiest either will get you the correct answer.