Question Video: Finding the Missing Vector in an Equation Involving Operations on Vectors Mathematics

If 𝐀 = ⟨−1, 1, 1⟩ and 𝐁 = ⟨1, 1, −2⟩, determine the vector 𝐂 for which 2𝐂 + 5𝐀 = 5𝐁.

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Video Transcript

If vector 𝐀 is equal to negative one, one, one and vector 𝐁 is equal to one, one, negative two, determine the vector 𝐂 for which two 𝐂 plus five 𝐀 is equal to five 𝐁.

In order to answer this question, we need to recall how to add and subtract vectors as well as multiply a vector by a scalar. We will begin though by rearranging the equation two 𝐂 plus five 𝐀 equals five 𝐁 to make vector 𝐂 the subject. We begin by subtracting five multiplied by vector 𝐀 from both sides. This gives us two 𝐂 is equal to five 𝐁 minus five 𝐀. Multiplying both sides of this equation by a half, we see that vector 𝐂 is equal to a half multiplied by five 𝐁 minus five 𝐀.

At this stage, we could distribute the parentheses by multiplying five 𝐁 minus five 𝐀 by one-half. However, in this question, we will calculate the vectors five 𝐁 and five 𝐀 first. To multiply any vector by a scalar, we simply multiply each of the individual components by that scalar. As five multiplied by one is equal to five and five multiplied by negative two is negative 10, five multiplied by vector 𝐁 is equal to five, five, negative 10. We can repeat this process to calculate five multiplied by vector 𝐀. This is equal to the vector negative five, five, five.

Our next step is to subtract these two vectors. We need to calculate five 𝐁 minus five 𝐀. To subtract any two vectors, we simply subtract their corresponding components. Five minus negative five is the same as five plus five, which equals 10. Five minus five is equal to zero. Finally, negative 10 minus five is equal to negative 15. Five 𝐁 minus five 𝐀 is equal to the vector 10, zero, negative 15.

Vector 𝐂 is therefore equal to a half multiplied by this vector. Once again, we can calculate this by multiplying each of the components by one-half. A half of 10 is equal to five. A half of zero is zero. And a half of negative 15 is negative 15 over two or negative fifteen-halves. The vector 𝐂 for which two 𝐂 plus five 𝐀 is equal to five 𝐁 is five, zero, negative 15 over two.

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