Video Transcript
A rectangle 𝐴𝐵𝐶𝐷 has vertices 𝐴 negative six, negative seven; 𝐵 zero, two; 𝐶 six, negative two; and 𝐷 zero, negative 11. Use vectors to determine its area.
We will begin by sketching the rectangle on a coordinate grid. Point 𝐴 has coordinates negative six, negative seven. Point 𝐵 has coordinates zero, two. Point 𝐶 is equal to six, negative two. And finally, point 𝐷 has coordinates zero, negative 11. We are asked to calculate the area of the rectangle using vectors. We know that the area of any parallelogram is equal to the magnitude of the cross product of vectors 𝐚 and 𝐛 where vector 𝐚 and vector 𝐛 are the sides of the parallelogram. The magnitude of the cross product of any two vectors is equal to the magnitude of vector 𝐚 multiplied by the magnitude of vector 𝐛 multiplied by the magnitude of sin 𝜃, where 𝜃 is the angle between the two vectors.
We know that a rectangle is a special type of parallelogram where the four angles are equal to 90 degrees. The area of rectangle 𝐴𝐵𝐶𝐷 is therefore equal to the magnitude of the cross product of vector 𝚨𝚩 and vector 𝚨𝐃. This, in turn, is equal to the magnitude of vector 𝚨𝚩 multiplied by the magnitude of vector 𝚨𝐃 multiplied by the magnitude of sin 𝜃. 𝜃 is equal to 90 degrees, and we know that the sin of 90 degrees is equal to one. The area of the rectangle is therefore equal to the magnitude of vector 𝚨𝚩 multiplied by the magnitude of vector 𝚨𝐃. We will now clear some space so we can calculate these two values.
Vector 𝚨𝚩 will have components equal to zero minus negative six and two minus negative seven. Zero minus negative six is equal to six and two minus negative seven is equal to nine. Therefore, vector 𝚨𝚩 is equal to six, nine. We can find the magnitude of any vector by finding the sum of the squares of each of the components and then square rooting the answer. Six squared is equal to 36, and nine squared is 81. Therefore, the magnitude of vector 𝚨𝚩 is the square root of 117. This simplifies to three root 13.
We can now repeat this process for vector 𝚨𝐃. This will have an 𝑥-component equal to zero minus negative six and a 𝑦-component equal to negative 11 minus negative seven. This is equal to six, negative four. The magnitude of vector 𝚨𝐃 is therefore equal to the square root of six squared plus negative four squared. As six squared is equal to 36 and negative four squared is equal to 16, we are left with the square root of 52. This simplifies to two root 13. Substituting these values into our equation gives us three root 13 multiplied by two root 13. Three multiplied by two is equal to six, and root 13 multiplied by root 13 is 13. This leaves us with six multiplied by 13, which is equal to 78. The area of rectangle 𝐴𝐵𝐶𝐷 is 78 area units.
