Question Video: Finding the Unknown Component of a Vector given the Result of Its Cross Product by Another Vector Mathematics

Video Transcript

If the vector 𝐀 is equal to three 𝐢 minus five 𝐣, 𝐁 is equal to 𝑚𝐢 plus five 𝐣, and the cross product of 𝐀 and 𝐁 is equal to 50𝐤, find the value of 𝑚.

To answer this question, we’re going to need to recall what we mean by the cross product of two vectors. Essentially, it’s a way of multiplying two vectors. Unlike the dot product, where we get a scalar quantity, when we find the cross product of two vectors, we get a vector. And we say that if 𝐚 is the three-dimensional vector given by 𝑎 one, 𝑎 two, 𝑎 three and 𝐛 is the vector 𝑏 one, 𝑏 two, 𝑏 three. Then the cross product of 𝐚 and 𝐛 is equal to the vector 𝑎 two 𝑏 three minus 𝑎 three 𝑏 two, 𝑎 three 𝑏 one minus 𝑎 one 𝑏 three, and 𝑎 one 𝑏 two minus 𝑎 two 𝑏 one.

Now, this isn’t a particularly easy formula to remember. And if you’re struggling, you might want to recall that it’s really just the determinant of a three-by-three matrix. So now, we have a formula. Let’s define 𝑎 one, 𝑎 two, 𝑎 three and 𝑏 one, 𝑏 two, 𝑏 three. 𝑎 one is the horizontal component for 𝐚. It’s three. 𝑎 two is the vertical component. That’s negative five. And there’s no 𝐤-component. So 𝑎 three is equal to zero. 𝑏 one is equal to 𝑚. 𝑏 two is equal to five. And once again, 𝑏 three is equal to zero.

The first element of the cross product of these two vectors then is 𝑎 two 𝑏 three. So that’s negative five times zero minus 𝑎 three 𝑏 two, which is zero multiplied by five. The second element is 𝑎 three 𝑏 one, which is zero times 𝑚, minus 𝑎 one 𝑏 three, which is three times zero. And the third element is 𝑎 one 𝑏 two, that’s three times five, minus 𝑎 two 𝑏 one, that’s negative five multiplied by 𝑚. The first and second elements simplify to zero. And the third element of the cross product of our two vectors becomes 15 plus five 𝑚. And that’s because we have minus negative five. So we get plus five.

Now, if we go back to the question, we see that the cross product of 𝐀 and 𝐁 is given to us. We’re told it’s 50𝐤. Well, another way to represent that is using these sort of triangular brackets. The vector 𝐚 crossed with 𝐛 is zero, zero, 50. And we see then that, for the vectors to be equal, 50 must be equal to 15 plus five 𝑚. We’ll solve by subtracting 15 from both sides. And that gives us 35 equals five 𝑚. We then divide both sides of this equation by five. And we get seven equals 𝑚.

And so if 𝐀 is equal to three 𝐢 minus five 𝐣, 𝐁 is equal to 𝑚𝐢 plus five 𝐣, and the cross product of 𝐀 and 𝐁 is 50𝐤, 𝑚 must be equal to seven.