Video: Application of Vectors in Space

Given 𝐀 = ⟨𝑎, −10, −9⟩ and 𝐁 = ⟨−3, 𝑏, 3⟩, if 𝐴𝐵 = ⟨5, 3, 𝑐⟩, find the value of 𝑎 + 𝑏 + 𝑐.

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Video Transcript

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.

We begin this question by recalling that 𝐴𝐵 is equal to vector 𝐁 minus vector 𝐀. We could rewrite each of the vectors in terms of 𝐢, 𝐣, and 𝐤. For example, vector 𝐀 is equal to 𝑎𝐢 minus 10𝐣 minus nine 𝐤. In this case, we will write them as column vectors, as shown. We can then solve each of the rows individually.

On the top row, we have the equation five equals negative three minus 𝑎. Adding three to both sides gives us eight is equal to negative 𝑎. Dividing both sides by negative one gives us a value of 𝑎 equal to negative eight. The second row gives us the equation three is equal to 𝑏 minus negative 10. Subtracting a negative number is the same as adding the absolute value of that number. So, the equation simplifies to three equals 𝑏 plus 10. We can then subtract 10 from both sides, giving us a value of 𝑏 equal to negative seven. Finally, the bottom row of the vectors gives us the equation 𝑐 is equal to three minus negative nine. This is the same as three plus nine. So, 𝑐 is equal to 12.

We have now calculated values of 𝑎, 𝑏, and 𝑐, which we can use to find the value of 𝑎 plus 𝑏 plus 𝑐. We need to add negative eight, negative seven, and 12. Negative eight plus negative seven is equal to negative 15. Adding 12 to this gives us negative three.

The value of 𝑎 plus 𝑏 plus 𝑐 equals negative three.

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