Determine whether the given vectors are orthogonal,...

Determine whether the given vectors are orthogonal, parallel, or neither.

(a) $ u = \langle -5, 4, -2 \rangle $ , $ v = \langle 3, 4, -1 \rangle $
(b) $ u = 9i - 6j + 3k $ , $ v = -6i + 4j - 2k $
(c) $ u = \langle c, c, c \rangle $ , $ v = \langle c, 0, -c \rangle $

Transcript

00:01 This problem, we have three pairs of vectors we're going to examine.

00:03 I've done one of them in black, one in red, and one in green.

00:07 And for each pair of vectors, we want to compare them.

00:10 And we want to know are they orthogonal, which means perpendicular to each other, are they parallel vectors, or neither.

00:22 So how can we tell what orientation these vectors are to each other? well, we have a theorem that's the cosine of theta, theta being the angle between two vectors.

00:32 The co -sign.

00:33 The co -sign of theta, the angle between two vectors, the cosine of theta is the dot product of the vectors divided by the product of the magnitudes of the vectors.

00:41 We can use this theorem to answer the question in this problem.

00:45 First of all, let's look at orthogonal.

00:48 If the vectors are parallel, that means that they're going to make a 90 degree angle.

00:52 So theta will be 90 degrees.

00:55 Well, cosine of 90 degrees is zero.

00:57 So the only way for this fraction in this theorem to equal to zero is if our numerator is equal to zero, which means that our dot product would have to equal zero.

01:08 So let's start there.

01:09 I'm going to look at each case and see if they're orthogonal.

01:12 Let's just do that first.

01:13 So i'm going to find the dot product of each of these pairs.

01:17 First let's start with the black one.

01:19 The dot product, u .d .v.

01:21 Well, remember, for a dot product, we're going to multiply the xes, multiply the ys, multiply the z, and we're going to add them up.

01:28 So for the first one, when i multiply the xes, i get negative 15, multiplying the y's gives me 16.

01:36 Multiplying the zs gives me two.

01:39 And that gives me a dot product of three.

01:41 So these are not orthogonal.

01:43 So i know that to start with.

01:45 Not orthogonal for the black.

01:48 What about my red pair? well, this dot product, when i multiply, in this case, because i'm giving it in i's, j's, and k, we still do the same thing.

01:58 I'm to multiply the coefficients for the i's, j's, and k separately, and add them up.

02:03 I get negative 54 minus 24 minus 6.

02:09 And that gives me a dot product of negative 84.

02:12 This also is not orthogonal.

02:16 What about the green ones? well, this dot product, when i multiply the x's, that gives me c squared.

02:23 The ys gives me zero...

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