Consider the two vectors 𝐩 equals four 𝐢 plus five 𝐣 and 𝐪 equals two 𝐢 plus eight 𝐣. Calculate 𝐩 cross 𝐪.
Alright, so in this question, we’re given two vectors in component form, 𝐩 and 𝐪. And we’re asked to calculate the vector product 𝐩 cross 𝐪. Let’s start by drawing a quick sketch of these vectors.
Notice that both 𝐩 and 𝐪 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 𝐩 and 𝐪 lie in the 𝑥𝑦-plane. Vector 𝐩 has four units in the 𝑥-direction and five units in the 𝑦-direction, meaning that the vector looks like this. Vector 𝐪 has two units in the 𝑥-direction and eight units in the 𝑦-direction, so it looks like this.
To answer the question, we need to evaluate the vector product 𝐩 cross 𝐪. So let’s recall our definition of the vector product of two vectors. 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 𝐀 and 𝐁. We can write the vectors in component form as 𝐀 equals an 𝑥-component, 𝐀 subscript 𝑥, multiplied by 𝐢 plus a 𝑦-component, 𝐀 subscript 𝑦, multiplied by 𝐣, and similarly for 𝐁. Then, the vector product 𝐀 cross 𝐁 is defined as the 𝑥-component of 𝐀, that’s 𝐀 subscript 𝑥, multiplied by the 𝑦-component of 𝐁, that’s 𝐁 subscript 𝑦, minus the 𝑦-component of 𝐀, 𝐀 subscript 𝑦, multiplied by the 𝑥-component of 𝐁, 𝐁 subscript 𝑥. And then this whole thing is multiplied by unit vector 𝐤, which points in the 𝑧-direction.
So the vector product 𝐀 cross 𝐁 produces a vector with this magnitude and with a direction that’s perpendicular to the direction of both 𝐀 and 𝐁. We can use this expression to calculate the vector product of the vectors 𝐩 and 𝐪 that we are given in the question. We are asked to work out 𝐩 cross 𝐪. So the first term in our vector product expression tells us that we need the 𝑥-component of 𝐩, which is four, multiplied by the 𝑦-component of 𝐪, which is eight. Then we subtract the second term from this. Again, looking at our expression for the vector product, we see that this second term tells us we need the 𝑦-component of vector 𝐩, which is five, multiplied by the 𝑥-component of 𝐪, which is two. Finally, we need to multiply this whole thing by the unit vector 𝐤.
All that remains now is to evaluate this expression here. If we do the multiplications, we get that the first term gives us 32 and the second term, which we subtract, gives us 10. Subtracting 10 from 32 gives us 22. And this gives us our answer to the question that the vector product 𝐩 cross 𝐪 is equal to 22𝐤.