Let 𝐴 equal one, one and 𝐵 equal one, three. Which of the following is the parametric form of the equation of line segment 𝐴𝐵? Is it (A) 𝑥 equals one, 𝑦 equals 𝑘 plus one? (B) 𝑥 equals one, 𝑦 equals two 𝑘 plus one. (C) 𝑥 equals one, 𝑦 equals two multiplied by 𝑘 plus one. (D) 𝑥 equals 𝑘 plus one, 𝑦 equals one. Or (E) 𝑥 equals two 𝑘 plus one, 𝑦 equals one.
We begin by recalling that when Cartesian coordinates of a curve are represented by a function of the same variable, usually 𝑡, they are called parametric equations. This means that parametric equations in the 𝑥𝑦-plane, 𝑥 equals 𝑓 of 𝑡 and 𝑦 equals 𝑔 of 𝑡, denote the 𝑥- and 𝑦-coordinates of the graph of a curve in the plane. In this case, the points 𝐴 and 𝐵 are as shown. 𝐴 has coordinates one, one and 𝐵 has coordinates one, three. We want the parametric equations of the line segment 𝐴𝐵.
We will begin by recalling how we find the vector equation of a straight line. The vector equation of a line that passes through the point with position vector 𝐫 sub zero and has direction vector 𝐝 is 𝐫 is equal to 𝐫 sub zero plus 𝑡 multiplied by 𝐝, where 𝑡, which is sometimes written as 𝑘 or 𝜇, is an unknown scalar or constant. We can choose any point that lies on the line to help find the position vector. In this case, we will choose point 𝐴 with coordinates one, one. This has position vector one, one and is therefore the value of 𝐫 sub zero.
The direction vector 𝐝 is equal to the vector 𝐀𝐁. And this is equal to vector 𝐎𝐁 minus vector 𝐎𝐀. Subtracting the position vector of point 𝐴 from the position vector of point 𝐵, we have one, three minus one, one. To subtract vectors, we simply subtract their corresponding components. This means that the direction vector 𝐀𝐁 is equal to zero, two.
Since the options in this question have used 𝑘 instead of 𝑡, we have the vector equation of the line is equal to one, one plus 𝑘 multiplied by zero, two. This can be simplified to one, one plus zero, two 𝑘. And adding our two vectors, we have 𝐫 is equal to one, two 𝑘 plus one.
Next, we rewrite the left-hand side as the vector 𝑥𝑦. Since the vector on the left-hand side of our equation must be equal to the vector on the right, the individual components must be equal. Therefore, 𝑥 is equal to one and 𝑦 is equal to two 𝑘 plus one. This is the parametric form of the equation of the line segment 𝐴𝐵. The correct answer is option (B) 𝑥 is equal to one and 𝑦 is equal to two 𝑘 plus one.