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Question Video: Solving a Vector Equation for a Scalar Multiple Mathematics

If <1, 2, 2> + π‘˜<2, 1, 2> = <5, 4, 6>, find π‘˜.

02:34

Video Transcript

If the vector one, two, two added to π‘˜ times the vector two, one, two is equal to the vector five, four, six, find π‘˜.

In this question, we’re given a vector equation involving a scalar π‘˜. We need to use this to determine the value of π‘˜. To answer this question, let’s start by recalling what it means for two vectors to be equal. We say that two vectors are equal if they have the same magnitude and direction. This is exactly the same as saying that the two vectors have equal dimension and all of their components are equal.

Therefore, we’ll start by simplifying the right-hand side of our equation to give us a single vector. First, to multiply a vector by a scalar, we just need to multiply all of the components of our vector by our scalar. So the vector π‘˜ multiplied by two, one, two is equal to the vector two π‘˜, one times π‘˜ β€” which simplifies to just π‘˜ β€” and then two π‘˜. So the right-hand side of this equation simplifies to give us the vector one, two, two added to the vector two π‘˜, π‘˜, two π‘˜.

We then recall that we can add two vectors of the same number of dimensions together by just adding the corresponding components together. Applying this to the two vectors on the right-hand side of our equation, we get the vector one plus two π‘˜, two plus π‘˜, two plus two π‘˜. And remember, we’re told that this is equal to the vector five, four, six. Since these two vectors are equal, their corresponding components must also be equal. For example, the first components of each of the two vectors must be equal. In other words, we must have that five is equal to one plus two π‘˜.

We can then solve this equation for the value of π‘˜. We do this first by subtracting one from both sides of the equation. This then gives us that four will be equal to two π‘˜. Next, we need to divide both sides of our equation through by two. This gives us that the value of π‘˜ is equal to two. But this just guarantees that the first component of our vectors are equal. We still need to check that the second components of the two vectors are equal and the third components of the two vectors are equal. We’ll do this by substituting π‘˜ is equal to two into our vector.

Substituting π‘˜ is equal to two on the vector on the right-hand side of our equation gives us the vector one plus two times two, two plus two, two plus two times two. And if we evaluate the expressions inside of each of our components, we get the vector five, four, six, which we can see is exactly equal to the left-hand side of this equation. Therefore, we’ve shown if the vector one, two, two added to π‘˜ times the vector two, one, two is equal to the vector five, four, six, then the value of π‘˜ must be equal to two.

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