Consider the two vectors 𝐀 equals six 𝐢 plus seven 𝐣 and 𝐁 equals 12𝐢 plus four 𝐣. Does 𝐀 cross 𝐁 point in the positive 𝑧-direction or the negative 𝑧-direction?
Alright, so this is a question about the vector product of two vectors 𝐀 and 𝐁. We are asked to work out whether the vector product 𝐀 cross 𝐁 points in the positive 𝑧-direction or the negative 𝑧-direction. If we look at these two vectors that we are given, we see that they both have an 𝐢-component and a 𝐣-component. Recall that 𝐢 is the unit vector in the 𝑥-direction and 𝐣 is the unit vector in the 𝑦-direction. This means that both our vectors 𝐀 and 𝐁 lie in the 𝑥𝑦-plane. We can draw these vectors as follows. Vector 𝐀 has six units in the 𝑥-direction and seven units in the 𝑦-direction, giving a vector like this. Vector 𝐁 has 12 units in the 𝑥-direction and four in the 𝑦-direction, so it looks like this.
To answer the question, we need to evaluate the vector product 𝐀 cross 𝐁. So let’s recall our definition of the vector product. Let’s consider two general vectors that lie in the 𝑥𝑦-plane. To distinguish them from the vectors that we’re given in the question, we’ll call these vectors lowercase 𝐚 and lowercase 𝐛. We can write the vectors in component form as 𝐚 equals 𝑎 subscript 𝑥 multiplied by 𝐢 plus 𝑎 subscript 𝑦 multiplied by 𝐣, and similarly for 𝐛. Then the vector product 𝐚 cross 𝐛 is defined as 𝑎 subscript 𝑥 multiplied by 𝑏 subscript 𝑦 minus 𝑎 subscript 𝑦 multiplied by 𝑏 subscript 𝑥 all multiplied by 𝐤, which is a unit vector in the positive 𝑧-direction. So the vector product 𝐚 cross 𝐛 produces a vector with this magnitude and with a direction that is perpendicular to the direction of both 𝐚 and 𝐛.
The question asks whether our particular vector product lies along positive 𝑧 or negative 𝑧. Looking at our general expression for the vector product, we see that this is equivalent to asking whether this bit here is positive or negative. If this term is positive, we have a positive number multiplied by a unit vector 𝐤 in the positive 𝑧-direction. Then, our vector resulting from calculating our vector product points in the positive 𝑧-direction. Meanwhile, if this term is negative, then we have a negative number multiplied by the unit vector in the positive 𝑧-direction. This gives us a vector that points in the negative 𝑧-direction. So let’s work out the vector product for the two vectors capital 𝐀 and capital 𝐁 that we’re given in the question.
The first term is the 𝑥-component of 𝐀, which is six, multiplied by the 𝑦-component of 𝐁, which is four. Then we subtract a second term from this. This second term is the 𝑦-component of 𝐀, which is seven, multiplied by the 𝑥-component of 𝐁, which is 12. Then this whole expression is multiplied by 𝐤, the unit vector in the positive 𝑧-direction. If we do the multiplications, we get that the first term in the brackets is 24 and the second term is 84. Subtracting 84 from 24 gives us negative 60. We therefore have a negative number multiplied by the unit vector in the positive 𝑧-direction. And, as we said before, this negative number means that our resulting vector points in the negative 𝑧-direction.
And so we have our answer to the question that when we calculate the vector product 𝐀 cross 𝐁, our resulting vector points in the negative 𝑧-direction.