Video Transcript
The diagram shows two vectors, 𝐀
and 𝐁, in three-dimensional space. Both vectors lie in the
𝑥𝑦-plane. Each of the squares on the grid has
a side length of one. Calculate 𝐀 cross 𝐁.
This question is asking us about
vector products. Specifically, we are asked to work
out the vector product 𝐀 cross 𝐁, where the vectors 𝐀 and 𝐁 are given to us in
the form of arrows drawn on a diagram, and we are told that both vectors lie in the
𝑥𝑦-plane.
Let’s begin by recalling the
definition of the vector product of two vectors. We’ll consider two general vectors,
𝐂 and 𝐃, and suppose that both of them lie in the 𝑥𝑦-plane. Then we can write these vectors in
component form as an 𝑥-component labeled with a subscript 𝑥 multiplied by 𝐢 hat
plus a 𝑦-component labeled with a subscript 𝑦 multiplied by 𝐣 hat. Remember that 𝐢 hat is the unit
vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction. Then the vector product 𝐂 cross 𝐃
is given by the 𝑥-component of 𝐂 multiplied by the 𝑦-component of 𝐃 minus the
𝑦-component of 𝐂 multiplied by the 𝑥-component of 𝐃. And this is all multiplied by 𝐤
hat, which is the unit vector in the 𝑧-direction.
This general expression for the
vector product is telling us that if we want to calculate 𝐀 cross 𝐁, then we’re
going to need to work out the 𝑥- and 𝑦-components of the vectors 𝐀 and 𝐁. The vectors 𝐀 and 𝐁 are both
shown in the diagram that’s given to us in the question. And the question also tells us that
each of the squares on the grid in this diagram has a side length of one. This means that in order to find
the 𝑥- and 𝑦-components of our vectors 𝐀 and 𝐁, all we need to do is count the
number of squares that each vector extends in the 𝑥-direction and the
𝑦-direction.
Let’s begin by doing this for
vector 𝐀. By tracing down from the tip of
vector 𝐀 to the 𝑥-axis, we can see that vector 𝐀 extends one, two, three, four
units in the negative 𝑥-direction. And by tracing across to the
𝑦-axis, we can see that 𝐀 extends one, two, three units in the positive
𝑦-direction. So we know that the 𝑥-component of
𝐀 is negative four and the 𝑦-component is positive three. In component form, we therefore
have that the vector 𝐀 is equal to negative four 𝐢 hat plus three 𝐣 hat.
Now let’s do the same thing with
vector 𝐁. If we trace down from the tip of
vector 𝐁 until it meets the 𝑥-axis, then we can see that 𝐁 extends one, two,
three, four squares into the positive 𝑥-direction. And if we trace across to the
𝑦-axis, we see that 𝐁 extends one, two, three, four, five squares into the
positive 𝑦-direction. So the 𝑥-component of 𝐁 is
positive four and the 𝑦-component is positive five. So writing vector 𝐁 in component
form, we have that 𝐁 is equal to four 𝐢 hat plus five 𝐣 hat.
We now have each of our vectors 𝐀
and 𝐁 written in component form, which means we are now ready to calculate the
vector product 𝐀 cross 𝐁. In our general expression for the
vector product of two vectors, we see that the first term is given by the
𝑥-component of the first vector in the product multiplied by the 𝑦-component of
the second vector in the product. In our case, the first vector in
our product is 𝐀 and the second vector is 𝐁. So we need the 𝑥-component of
vector 𝐀, which is negative four, multiplied by the 𝑦-component of vector 𝐁,
which is five. We then subtract a second term from
this first one.
This second term is the
𝑦-component of the first vector in the product multiplied by the 𝑥-component of
the second vector in the product. So for us, that’s the 𝑦-component
of vector 𝐀, which is three, multiplied by the 𝑥-component of vector 𝐁, which is
four. And then this whole thing gets
multiplied by the unit vector 𝐤 hat.
The final step is to evaluate this
expression here. The first term, negative four
multiplied by five, gives us negative 20. And the second term, three
multiplied by four, gives us 12. When we calculate negative 20 minus
12, we get a value of negative 32. And so our answer to the question
is that the vector product 𝐀 cross 𝐁 is equal to negative 32𝐤 hat.
