Question Video: Calculating a 3d Cross Product Using Vectors Defined by a Rectangle Mathematics

Video Transcript

In the rectangle 𝐴𝐵𝐶𝐷 shown in the figure, calculate 𝐃𝐀 cross 𝐁𝐌 if 𝐢, 𝐣, and 𝐤 form a right-hand system of unit vectors.

Looking at our rectangle, we see it’s eight centimeters tall and 16 centimeters wide and it exists in a space defined by a right-hand system of unit vectors 𝐢, 𝐣, and 𝐤. Relative to the rectangle, we see the 𝐢 hat unit vector points to the right and the 𝐣 hat unit vector points up. This tells us that the 𝐤 hat unit vector points out of the screen at us. Anyway, we want to calculate this cross product, 𝐃𝐀 cross 𝐁𝐌. Let’s first define these vectors. Vector 𝐃𝐀 is a vector from point 𝐷 to point 𝐴 in our rectangle. And likewise, vector 𝐁𝐌 is a vector from point 𝐵 to point 𝑀, the middle of our rectangle.

Now because this rectangle lies in what we can call the 𝐢𝐣-plane and because we can say that one centimeter on this plane is equal to one unit of distance, we can then write these vectors 𝐃𝐀 and 𝐁𝐌 according to their 𝐢 hat and 𝐣 hat components. Considering first vector 𝐃𝐀, we see that it’s entirely in the vertical direction. That means it will have no 𝐢 hat component. It will, however, have a negative 𝐣 hat component. And the reason is that this vector points opposite the positive 𝐣 hat direction.

That distance we know, the length of this vector, equals the length of the height of our rectangle, eight centimeters or just eight. Keeping both 𝐢 hat and 𝐣 hat components then, 𝐃𝐀 equals zero 𝐢 hat minus eight 𝐣 hat. Looking next at vector 𝐁𝐌, we see that this vector will have both an 𝐢 hat and a 𝐣 hat component because it points diagonally. It’s the components of this vector that we’re after, in other words, this vertical distance here and this horizontal distance here.

Considering its horizontal component, we know that vector 𝐁 points to the left. That’s in the negative 𝐢 hat direction. And since it goes to the midpoint of our rectangle — which is 16 centimeters wide — it will have an 𝐢 hat component of negative eight. Considering then the 𝐣 hat component — this part of our vector — we see that it points in the positive 𝐣 hat direction and that it’s equal in length to one-half the height of our rectangle. The 𝐣-component of 𝐁𝐌 then is positive four.

So we’ve now got our two vectors defined in terms of the unit vectors of our space. We can now move towards calculating their cross product. In general, if we have two vectors — we’ll call them 𝐀 and 𝐁 — that lie in the 𝐢 hat and 𝐣 hat plane, then the cross product 𝐀 cross 𝐁 equals the 𝐢-component of 𝐀 times the 𝐣-component of 𝐁 minus the 𝐣-component of 𝐀 times the 𝐢-component of 𝐁 all in the 𝐤 hat unit vector direction.

Notice then that this cross product is perpendicular to both vectors 𝐀 and 𝐁. And this is always the case for a cross product. When we go to cross 𝐃𝐀 and 𝐁𝐌 then, our formula tells us that this equals the 𝐢-component of our first vector — that’s vector 𝐃𝐀 and that 𝐢-component is zero — multiplied by the 𝐣-component of our second vector — that second vector is 𝐁𝐌 and the 𝐣-component is four. Then from this we subtract the 𝐣-component of our first vector — that vector is 𝐃𝐀 and its 𝐣-component is negative eight — multiplied by the 𝐢-component of our second vector — that second vector is 𝐁𝐌 and its 𝐢-component is negative eight. And this resulting vector, we know, is in the 𝐤 hat unit vector direction.

Evaluating this result, we know that zero times four is zero and negative eight times negative eight is positive 64. Our overall outcome then is negative 64𝐤 hat. 𝐃𝐀 cross 𝐁𝐌 then results in a vector that points 64 units or 64 centimeters into the screen.