Question Video: Solving for Vector Components Using Parallel and Perpendicular Vector Relationships Mathematics

If 𝚨 = ⟨12, 8⟩, 𝚩 = ⟨3, π‘šβŸ©, 𝐂 = βŸ¨π‘›, 9⟩, and 𝚨 βˆ₯ 𝚩, 𝚩 βŠ₯ 𝐂, then π‘š + 𝑛 = οΌΏ.

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Video Transcript

If 𝚨 equals 12, eight; 𝚩 equals three, π‘š; 𝐂 equals 𝑛, nine; and 𝚨 is parallel to 𝚩, 𝚩 is perpendicular to 𝐂, then π‘š plus 𝑛 equals what.

So, here we have these three vectors 𝚨, 𝚩, and 𝐂, and we’re told that 𝚨 and 𝚩 are parallel while 𝚩 and 𝐂 are perpendicular. As a side note, this means that 𝚨 is also perpendicular to 𝐂. But we won’t need to use that fact to solve for π‘š and 𝑛, these components of vectors 𝚩 and 𝐂, respectively. We can start out by solving for the value of π‘š in vector 𝚩. Now, the fact that this vector is parallel to vector 𝚨 means that we can write, for some nonzero constant 𝐢, that 𝐢 times 𝚩 equals 𝚨. Another way of saying this is that the π‘₯-component of 𝐴 over the π‘₯-component of 𝐡 equals that same ratio for the corresponding 𝑦-components of these vectors.

From the given information, we see that 𝐴 sub π‘₯ equals 12, 𝐡 sub π‘₯ equals three, 𝐴 sub 𝑦 equals eight, while 𝐡 sub 𝑦 is π‘š. So, 12 over three equals eight over π‘š. And if we multiply both sides of this equation by three and by π‘š, we find that 12 times π‘š equals three times eight or 24. Dividing both sides of this equation by 12, we find that π‘š equals two. So, we now know part of our answer of what π‘š plus 𝑛 will be. We can now use this information and the fact that 𝚩 is perpendicular to vector 𝐂 to solve for 𝑛.

In general, if two vectors, we’ll call them 𝐕 one and 𝐕 two, are perpendicular to one another, that means their dot product is zero. So, since 𝚩 is perpendicular to 𝐂, we can write that 𝚩 dot 𝐂 equals zero, and therefore three, π‘š dot 𝑛, nine equals zero. Before we calculate this dot product, let’s substitute in our known value for π‘š. We know π‘š is two, so three, two dot 𝑛, nine equals zero. And now, carrying out this dot product by first multiplying corresponding components of these vectors, we find that three times 𝑛 plus two times nine is zero so that subtracting 18 from both sides, we have that three times 𝑛 equals negative 18. Dividing both sides by three, we find that 𝑛 equals negative six. We have then our values of π‘š and 𝑛, and adding them together, we get a result of negative four.

So, if 𝚨 equals 12, eight; 𝚩 equals three, π‘š; 𝐂 equals 𝑛, nine; and 𝚨 is parallel to 𝚩, 𝚩 is perpendicular to 𝐂, then π‘š plus 𝑛 equals negative four.

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