In the figure, if 𝐴𝐵𝐶𝐷 is a parallelogram, then vector 𝐁𝐀 plus vector 𝐁𝐂 is equal to blank. Is it (A) vector 𝐀𝐂, (B) two multiplied by vector 𝐀𝐌, (C) vector 𝐃𝐁, or (D) two multiplied by vector 𝐌𝐃?
We recall that two vectors are equal if they have the same magnitude and direction. This means that in the parallelogram, the parallel sides will be represented by the same vectors as they have the same magnitude and direction. This means that the vector 𝐁𝐀 will be equal to the vector 𝐂𝐃 and the vector 𝐁𝐂 will be equal to the vector 𝐀𝐃. In this question, we’re given the expression vector 𝐁𝐀 plus vector 𝐁𝐂. We need to work out which of the four options is equivalent to this.
As already mentioned, vector 𝐁𝐀 is equal to vector 𝐂𝐃. This means we can rewrite our expression as vector 𝐂𝐃 plus vector 𝐁𝐂. Vector addition is commutative, so this can be rewritten as vector 𝐁𝐂 plus vector 𝐂𝐃. We notice that the endpoint of our first vector is the same as the start point of our second vector. This means that the vector 𝐁𝐂 plus the vector 𝐂𝐃 is equal to the vector 𝐁𝐃.
The diagonals of a parallelogram intersect at their midpoint. This means that the vector 𝐁𝐌 must be equal to the vector 𝐌𝐃. These will both be equal to one-half multiplied by vector 𝐁𝐃. If 𝐌𝐃 is equal to a half of 𝐁𝐃, then 𝐁𝐃 is equal to two multiplied by 𝐌𝐃, as we can multiply both sides of the equation by two.
This means that the correct answer is option (D). If 𝐴𝐵𝐶𝐷 is a parallelogram, then vector 𝐁𝐀 plus vector 𝐁𝐂 is equal to two multiplied by vector 𝐌𝐃.