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Question Video: Using Vectors to Find the Coordinates of a Vertex in a Rectangle Mathematics

𝐴𝐡𝐢𝐷 is a rectangle in which the coordinates of the points 𝐴, 𝐡, and 𝐢 are (βˆ’18, βˆ’2), (βˆ’18, βˆ’3), and (βˆ’8, π‘˜), respectively. Use vectors to find the value of π‘˜ and the coordinates of point 𝐷.

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Video Transcript

𝐴𝐡𝐢𝐷 is a rectangle in which the coordinates of the points 𝐴, 𝐡, and 𝐢 are negative 18, negative two; negative 18, negative three; and negative eight, π‘˜, respectively. Use vectors to find the value of π‘˜ 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, π‘˜. This is equal to negative 10, negative three minus π‘˜.

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 π‘˜ is equal to zero. This simplifies to zero is equal to three plus π‘˜. Subtracting three from both sides of this equation gives us π‘˜ is equal to negative three. The value of π‘˜ is equal to negative three, which means that 𝐢 has coordinates negative eight, negative three.

If we let the coordinates of point 𝐷 be π‘₯, 𝑦, then vector 𝐃𝐂 is equal to negative eight, negative three minus π‘₯, 𝑦. This is equal to negative eight minus π‘₯, 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 π‘₯. Negative one must be equal to negative three minus 𝑦. Solving our first equation, we get π‘₯ 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.

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