WEBVTT
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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.
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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.
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We will begin though by rearranging the equation two 𝐂 plus five 𝐀 equals five 𝐁 to make vector 𝐂 the subject.
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We begin by subtracting five multiplied by vector 𝐀 from both sides.
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This gives us two 𝐂 is equal to five 𝐁 minus five 𝐀.
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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.
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At this stage, we could distribute the parentheses by multiplying five 𝐁 minus five 𝐀 by one-half.
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However, in this question, we will calculate the vectors five 𝐁 and five 𝐀 first.
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To multiply any vector by a scalar, we simply multiply each of the individual components by that scalar.
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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.
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We can repeat this process to calculate five multiplied by vector 𝐀.
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This is equal to the vector negative five, five, five.
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Our next step is to subtract these two vectors.
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We need to calculate five 𝐁 minus five 𝐀.
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To subtract any two vectors, we simply subtract their corresponding components.
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Five minus negative five is the same as five plus five, which equals 10.
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Five minus five is equal to zero.
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Finally, negative 10 minus five is equal to negative 15.
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Five 𝐁 minus five 𝐀 is equal to the vector 10, zero, negative 15.
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Vector 𝐂 is therefore equal to a half multiplied by this vector.
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Once again, we can calculate this by multiplying each of the components by one-half.
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A half of 10 is equal to five.
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A half of zero is zero.
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And a half of negative 15 is negative 15 over two or negative fifteen-halves.
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The vector 𝐂 for which two 𝐂 plus five 𝐀 is equal to five 𝐁 is five, zero, negative 15 over two.