Question Video: Determining the Vector That is Not Perpendicular to the Given Line Mathematics

Which of the following vectors is not perpendicular to the line whose direction vector 𝐫 is ⟨2, βˆ’3, 5⟩? [A] ⟨10, 10, 2⟩ [B] βŸ¨βˆ’10, βˆ’5, 1⟩ [C] ⟨2, βˆ’2, βˆ’2⟩ [D] ⟨1, βˆ’2, 3⟩ [E] ⟨2, 3, 1⟩

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Video Transcript

Which of the following vectors is not perpendicular to the line whose direction vector, vector 𝐫, is two, negative three, five? Option (A) vector 10, 10, two. Option (B) vector negative 10, negative five, one. Option (C) vector two, negative two, negative two. Option (D) vector one, negative two, three. Or option (E) vector two, three, one.

The first thing we should recall is that if two vectors 𝐀 and 𝐁 are perpendicular to one another, then it must be true that their dot product is equal to zero. And so, in order to find which of these vectors is not perpendicular, then we need to find the vector which does not give zero when dotted with the vector two, negative three, five. In order to actually work out the dot product, we can recall that if a vector 𝐀 is given as 𝐀 sub π‘₯, 𝐀 sub 𝑦, and 𝐀 sub 𝑧 and vector 𝐁 is given as 𝐁 sub π‘₯, 𝐁 sub 𝑦, 𝐁 sub 𝑧, then vector 𝐀 dot vector 𝐁 is equal to 𝐀 sub π‘₯ times 𝐁 sub π‘₯ plus 𝐀 sub 𝑦 times 𝐁 sub 𝑦 plus 𝐀 sub 𝑧 times 𝐁 sub 𝑧.

So let’s begin with option (A), and we’ll take the dot product of the vector 10, 10, two and the vector two, negative three, five. And so this will be equal to 10 times two plus 10 times negative three plus two times five. 10 times two is 20, 10 times negative three is negative 30, and two times five is 10. And when we simplify 20 minus 30 plus 10, we get an answer of zero. Since the dot product of this vector given in option (A) and the vector two, negative three, five is equal to zero, then that means that these two vectors are perpendicular.

We are looking for a vector which is not perpendicular to the given vector. So let’s have a look at the vector given an option (B). Once again, we find the dot product of this vector negative 10, negative five, one and the vector two, negative three, five. This gives us negative 10 times two plus negative five times negative three plus one times five. We can simplify this to negative 20 plus 15 plus five, which gives us an answer of zero. Once again, we have found another perpendicular vector.

So let’s try option (C). The dot product of vector two, negative two, negative two and vector two, negative three, five is two times two plus negative two times negative three plus negative two times five. Simplifying this answer, we get a value of zero. And we have found a third perpendicular vector to the vector 𝐫. However, when we look at option (D) and we take the dot product of this vector and the vector two, negative three, five, we get an answer of 23. Since the dot product of these two vectors is not equal to zero, then we can say that these two vectors are not perpendicular.

It looks as though we have found the answer, but it’s worthwhile checking option (E) just to be certain. And when we take the dot product of the vector two, three, one and the vector two, negative three, five, we do indeed get an answer of zero. These two vectors would be perpendicular. And so we can give the answer that the vector which is not perpendicular to the line with direction vector two, negative three, five is that given in option (D). It’s the vector one, negative two, three.

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