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Given vector 𝐀 is equal to 𝑎, negative 10, negative nine and 𝐁 is equal to negative three, 𝑏, three, if 𝐴𝐵 equals five, three, 𝑐, find the value of 𝑎 plus 𝑏 plus 𝑐.
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We begin this question by recalling that 𝐴𝐵 is equal to vector 𝐁 minus vector 𝐀.
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We could rewrite each of the vectors in terms of 𝐢, 𝐣, and 𝐤.
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For example, vector 𝐀 is equal to 𝑎𝐢 minus 10𝐣 minus nine 𝐤.
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In this case, we will write them as column vectors, as shown.
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We can then solve each of the rows individually.
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On the top row, we have the equation five equals negative three minus 𝑎.
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Adding three to both sides gives us eight is equal to negative 𝑎.
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Dividing both sides by negative one gives us a value of 𝑎 equal to negative eight.
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The second row gives us the equation three is equal to 𝑏 minus negative 10.
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Subtracting a negative number is the same as adding the absolute value of that number.
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So, the equation simplifies to three equals 𝑏 plus 10.
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We can then subtract 10 from both sides, giving us a value of 𝑏 equal to negative seven.
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Finally, the bottom row of the vectors gives us the equation 𝑐 is equal to three minus negative nine.
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This is the same as three plus nine.
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So, 𝑐 is equal to 12.
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We have now calculated values of 𝑎, 𝑏, and 𝑐, which we can use to find the value of 𝑎 plus 𝑏 plus 𝑐.
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We need to add negative eight, negative seven, and 12.
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Negative eight plus negative seven is equal to negative 15.
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Adding 12 to this gives us negative three.
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The value of 𝑎 plus 𝑏 plus 𝑐 equals negative three.