Given the information in the diagram below, if 𝐁𝐃 equals 𝑘 times 𝐃𝐀, find 𝑘.
In our diagram, we see that the bottom side of this triangle is parallel to the line segment that runs from point 𝐸 to point 𝐷. Our problem statement tells us that this vector here, from point 𝐵 to point 𝐷, is equal to some constant 𝑘 times the vector from 𝐷 to 𝐴. It’s 𝑘 we want to solve for. And to do that, let’s rearrange this equation so 𝑘 is the subject.
Dividing both sides by vector 𝐃𝐀, this cancels on the right. And we see then that 𝑘 equals 𝐁𝐃 over 𝐃𝐀. Because these two vectors point in the same direction, we know that as we solve for this constant value 𝑘, it’s really the magnitudes or the lengths of these two vectors that we’re comparing. This means we can add vertical bars indicating magnitudes around the vectors of this expression. And from here we just read off these values from our diagram.
Vector 𝐁𝐃 has a magnitude of 4.8 centimeters, and 𝐃𝐀 has a magnitude of 7.2 centimeters. Note that in this fraction, the units will cancel out. And 4.8 divided by 7.2 simplifies to two-thirds. This is the value by which we can multiply vector 𝐃𝐀 so that it’s equal to vector 𝐁𝐃.