Question

If u = ❬6, –9❭ and v = ❬–24, 36❭ with an angle θ between the vectors, are u and v parallel or orthogonal? Explain.

The vectors are parallel because cos θ = −1.
The vectors are parallel because u • v = 0.
The vectors are orthogonal because u • v = 0.
The vectors are orthogonal because cos θ = −1.

Answer 1

The vectors u = ❬6, –9❭ and v = ❬–24, 36❭ are parallel to each other because the cosine of the angle θ is equal to -1. (Right choice: A)

How to determine the relationship between two vectors

By linear algebra we can use the definition of dot product to determine the relationship between two vectors, whose cases are described below:

1. Two vectors are orthogonal if and only if cos θ = 0.
2. Two vectors are parallel if and only if cos θ = 1 or cos θ = -1.

And the expression for the cosine of the angle between the two vectors is:

$\cos \theta = \frac{\vec u \,\bullet\,\vec v}{\|\vec u\|\cdot \|\vec v\|}$     (1)

Where $\|\vec u\|$ and $\|\vec v\|$ are the norms of the two vectors, which are determined by Pythagorean theorem.

Now we proceed to calculate the angle between the two vectors:

$\cos \theta = \frac{(6)\cdot (-24)+(-9)\cdot (36)}{(3\sqrt{13})\cdot (12\sqrt{13})}$

cos θ = -1

Thus, the vectors u = ❬6, –9❭ and v = ❬–24, 36❭ are parallel to each other because the cosine of the angle θ is equal to -1. (Right choice: A)

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