Video Transcript
𝐴𝐵𝐶𝐷 is a rectangle in which
the coordinates of the points 𝐴, 𝐵, and 𝐶 are negative 18, negative two; negative
18, negative three; and negative eight, 𝑘, respectively. Use vectors to find the value of 𝑘
and the coordinates of point 𝐷.
One way of solving this problem
would be to draw a rectangle on the coordinate plane; however, we are asked to use
vectors. It therefore makes sense to
consider some of the properties of a rectangle. A rectangle has two pairs of
equal-length parallel sides. This means that the vector 𝐀𝐁
will be equal to the vector 𝐃𝐂. Likewise, the vector 𝐃𝐀 will be
equal to the vector 𝐂𝐁. We also know that the angles in a
rectangle are right angles. This means that vector 𝐀𝐁 is
perpendicular to vector 𝐂𝐁. The same is true for the other
sides that meet at right angles.
We recall that to calculate vector
𝐀𝐁, we subtract vector 𝐀 from vector 𝐁. In this question, vector 𝐀𝐁 is
equal to negative 18, negative three minus negative 18, negative two. Negative 18 minus negative 18 is
the same as negative 18 plus 18. This is equal to zero. Negative three minus negative two
is equal to negative one. Therefore, vector 𝐀𝐁 equals zero,
negative one. Vector 𝐂𝐁 is equal to vector 𝐁
minus vector 𝐂. This is equal to negative 18,
negative three minus negative eight, 𝑘. This is equal to negative 10,
negative three minus 𝑘.
We know that if two vectors are
perpendicular, the scalar product equals zero. This means that the scalar product
of 𝐀𝐁 and 𝐂𝐁 equals zero. Zero multiplied by negative 10 plus
negative one multiplied by negative three minus 𝑘 is equal to zero. This simplifies to zero is equal to
three plus 𝑘. Subtracting three from both sides
of this equation gives us 𝑘 is equal to negative three. The value of 𝑘 is equal to
negative three, which means that 𝐶 has coordinates negative eight, negative
three.
If we let the coordinates of point
𝐷 be 𝑥, 𝑦, then vector 𝐃𝐂 is equal to negative eight, negative three minus 𝑥,
𝑦. This is equal to negative eight
minus 𝑥, negative three minus 𝑦. As the vectors 𝐀𝐁 and 𝐃𝐂 have
the same magnitude and direction, they must be equal. This means zero must be equal to
negative eight minus 𝑥. Negative one must be equal to
negative three minus 𝑦. Solving our first equation, we get
𝑥 is equal to negative eight. And solving the second equation, we
get 𝑦 is equal to negative two. The coordinates of point 𝐷 are
therefore equal to negative eight, negative two.
