Question Video: Finding a Vector given the Results of Its Product by Two Given Vectors Mathematics

Video Transcript

If vector 𝐀 is equal to negative 𝐢 minus two 𝐣, vector 𝐁 is equal to negative four 𝐢 minus four 𝐣, the cross product of vector 𝐀 and 𝐂 is negative three 𝐤, and the cross product of vector 𝐂 and vector 𝐁 is four 𝐤, find vector 𝐂.

In this question, we’re given vectors 𝐀 and 𝐁 which lie in the coordinate plane with 𝐢 and 𝐣 as unit vectors. We are also given the cross product of vector 𝐀 and 𝐂 together with the cross product of vector 𝐂 and 𝐁. And we are asked to find vector 𝐂.

We will let this vector be 𝑥𝐢 plus 𝑦𝐣, where 𝑥 and 𝑦 are constants. We recall that if we have two vectors 𝐦 and 𝐧 such that 𝐦 is equal to 𝑒𝐢 plus 𝑓𝐣 and 𝐧 is equal to 𝑔𝐢 plus ℎ𝐣, then the cross product of vectors 𝐦 and 𝐧 is equal to the determinant of the two-by-two matrix 𝑒, 𝑓, 𝑔, ℎ multiplied by the unit vector 𝐤, where the unit vector 𝐤 is perpendicular to the 𝑥𝑦-plane.

Let’s begin by considering the cross product of vectors 𝐀 and 𝐂. Using our general rule, this must be equal to the determinant of the two-by-two matrix negative one, negative two, 𝑥, 𝑦 multiplied by the unit vector 𝐤. The determinant of a matrix is equal to negative one multiplied by 𝑦 minus 𝑥 multiplied by negative two. This is equal to negative 𝑦 plus two 𝑥. So the cross product of vector 𝐀 and vector 𝐂 is negative 𝑦 plus two 𝑥 multiplied by the unit vector 𝐤. We already know that this is equal to negative three 𝐤. We can therefore conclude that negative three is equal to negative 𝑦 plus two 𝑥.

Adding 𝑦 and three to both sides of this equation, we have 𝑦 is equal to two 𝑥 plus three. We will call this equation one and now repeat the process for the cross product of vector 𝐂 and 𝐁. This is equal to the determinant of the two-by-two matrix 𝑥, 𝑦, negative four, negative four multiplied by 𝐤. The determinant of the matrix is 𝑥 multiplied by negative four minus negative four multiplied by 𝑦. This simplifies to negative four 𝑥 plus four 𝑦 such that the cross product of 𝐂 and 𝐁 is negative four 𝑥 plus four 𝑦 multiplied by the unit vector 𝐤. We are told that this is equal to four 𝐤. And negative four 𝑥 plus four 𝑦 is therefore equal to four.

We can divide both sides of this equation by four such that one is equal to negative 𝑥 plus 𝑦. And adding 𝑥 to both sides, we have 𝑦 is equal to 𝑥 plus one. We now have a pair of simultaneous equations we can solve to calculate the values of 𝑥 and 𝑦. Since 𝑦 is equal to two 𝑥 plus three and 𝑥 plus one, then these two expressions must be equal to one another. We can then subtract 𝑥 and three from both sides of the equation. Two 𝑥 minus 𝑥 is 𝑥, and one minus three is negative two, giving us 𝑥 is equal to negative two. We can then substitute this value of 𝑥 into equation one or equation two.

In Equation two, we have 𝑦 is equal to negative two plus one, which gives us 𝑦 is equal to negative one. Vector 𝐂 is therefore equal to negative two 𝐢 minus 𝐣. If 𝐀 is equal to negative 𝐢 minus two 𝐣, 𝐁 is equal to negative four 𝐣 minus four 𝐣, the cross product of 𝐀 and 𝐂 is negative three 𝐤, and the cross product of 𝐂 and 𝐁 is four 𝐤, then vector 𝐂 is equal to negative two 𝐢 minus 𝐣.