Video Transcript
Given that 𝐀 equals negative three, negative five, 𝐁 is parallel to 𝐀, and the magnitude of 𝐁 equals four times the square root of 34, find 𝐁.
In this question, we’re given a vector 𝐀 in its component form. And we’re also told about a vector 𝐁 whose magnitude we know, but we don’t yet know its components. We are told, though, that these two vectors are parallel. That being the case, it means that there exists some nonzero constant, we’ll call it 𝐶, by which we can multiply vector 𝐀 so it equals vector 𝐁. This statement is true because the two vectors are parallel. If we can solve for 𝐶 then, we can use our knowledge of vector 𝐀 to solve for the components of vector 𝐁.
We can start by calculating the magnitude of vector 𝐀. This equals the square root of the sum of the squares of its components. Negative three squared is nine and negative five squared is 25. So the magnitude of 𝐀 equals the square root of 34. Now, we can compare this to the magnitude of 𝐁. Since the magnitude of 𝐁 is four times the square root, we can write that the magnitude of 𝐁 equals four times the magnitude of 𝐀. This means the overall length of vector 𝐁 is four times as long as the overall length of vector 𝐀. And not only that, but because these vectors are parallel, we can say that each component of vector 𝐁 is four times longer than the corresponding component of vector 𝐀.
If we didn’t know that 𝐁 and 𝐀 are parallel, we couldn’t make this claim. But because they are, we can say that four is equal to 𝐶 in this equation. That’s the mathematical statement of saying that each component of vector 𝐁 is four times as long as that of vector 𝐀. So this constant of proportionality 𝐶 is four. And from our problem statement, we know that 𝐀 is negative three, negative five. Vector 𝐁 then is equal to negative 12, negative 20. And these are the components of vector 𝐁, which is parallel to vector 𝐀.
