Find the value(s) of a making v = 7ai

Find the value(s) of a making v = 7ai - 3j parallel to w = a^2 i + 6j. Remember that two vectors are parallel if they are scalar multiples of one another, i.e. v is parallel to w if there exists a scalar ? such that v = ? w. a = (If there is more than one value of a, enter the values as a comma-separated list.)

Transcript

00:01 The vectors v equal to 7a in the x direction and negative 3 in the y direction or in the i hat direction and then the j hat direction.

00:15 And then we have another vector w.

00:18 We have another vector w that will be a little bit weird.

00:23 It'll be a squared in the x direction and then positive 6 in the y direction.

00:30 So if two vectors are parallel, let's say v is parallel to w, then i can write both of these relations like this.

00:37 V is equal to some constant.

00:40 This will be a scalar.

00:42 Let's call it a scalar instead.

00:45 Some scalar times the same vector.

00:47 Now if we look at the components, which is why i kind of wrote it like this, if we look at the components, this has to be the same scalar across the vector.

00:54 So it already looks like i'm multiplying v1 by negative 2 or v, i guess the vector is by negative 2.

01:04 All right.

01:07 So that would give me negative 14a and positive 6.

01:17 So i'm going to say my, i'm going to cautiously say my scalar is negative 2.

01:22 Now if that's the case, that means that these two need to be equal as well.

01:28 So what we have is a squared is equal to 14a, negative 14a, excuse me, because obviously 6 is equal to 6.

01:40 So if we can equate these with a scalar multiple somehow, then they can be said to be equal, the components at least.

01:47 So a squared is equal to 14a.

01:50 This is quadratic, so we need to move everything to one side.

01:52 A squared plus 14a is equal to 0.

01:59 So one option, of course, is that a could be equal to 0...

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