Given that the vector 𝐀 is equal to negative two, seven and 𝐁 is equal to three, negative eight, determine the area of the parallelogram whose adjacent sides are represented by 𝐀 and 𝐁.
We begin by recalling that we can use the cross product to find the area of a parallelogram. It’s given by the magnitude of the cross product. But what do we mean by the cross product of two vectors? Well, the cross product is just a way of multiplying two vectors. It’s quite different to the dot product in that the dot product results in a scalar whereas a cross product results in another vector.
So given two vectors 𝑎 equals 𝑎 one, 𝑎 two, 𝑎 three and 𝑏 equals 𝑏 one, 𝑏 two, 𝑏 three, the cross product of these two vectors is 𝑎 two 𝑏 three minus 𝑎 three 𝑏 two, 𝑎 three 𝑏 one minus 𝑎 one 𝑏 three, and 𝑎 one 𝑏 two minus 𝑎 two 𝑏 one. So let’s begin by finding the cross product of our vector 𝐀 and 𝐁. We define 𝑎 one to be equal to negative two. That’s the first element in this vector. 𝑎 two is equal to seven. That’s the second element. And then, the third element is in fact zero. 𝑏 one is the first element in the vector 𝐁. It’s three. 𝑏 two is negative eight. And 𝑏 three is also zero.
Then, 𝑎 two 𝑏 three minus 𝑎 three 𝑏 two is seven times zero minus zero times negative eight. Then, 𝑎 three 𝑏 one minus 𝑎 one 𝑏 three is zero multiplied by three minus negative two multiplied by zero. Finally, we have 𝑎 one 𝑏 two minus 𝑎 two 𝑏 one. So that’s negative two multiplied by negative eight minus seven multiplied by three. The first two elements in our cross product are zero. Then, the third element becomes 16 minus 21, which is negative five.
Now, of course, we’re finding the magnitude of the cross product. So we need these absolute value bars. And you might be able to spot what the magnitude of the cross product of 𝐀 and 𝐁 is. But if not, let’s use the formula. It’s the square root of the sum of the squares of each of the individual components. So that’s the square root of zero squared plus zero squared plus negative five squared. Zero squared plus zero squared plus negative five squared is 25. And so the magnitude of the cross product is the square root of 25, which is five.
Since the area of the parallelogram is equal to the magnitude of the cross product, we can say that the area of the parallelogram we have is five square units.