Video Transcript
Fill in the blank. The direction vector of the straight line whose parametric equations are 𝑥 is equal to five 𝑘 plus two and 𝑦 is equal to negative three is what.
In this question, we’re given a pair of parametric equations which represent a straight line. 𝑥 is equal to five 𝑘 plus two, and 𝑦 is equal to negative three. We need to use these parametric equations to determine a direction vector of this line. To do this, we can start by recalling a direction vector of a line is a nonzero vector which runs parallel to the line. We can also recall exactly what is meant by the parametric equations of a line. It’s a pair of equations of the form 𝑥 is equal to 𝑥 sub zero plus 𝑎 times 𝑘 and 𝑦 is equal to 𝑦 sub zero plus 𝑏 multiplied by 𝑘, where the point with coordinates 𝑥 sub zero, 𝑦 sub zero is any point which lies on the line and the vector 𝐚, 𝐛 is a nonzero vector which is parallel to the line.
For example, we can look at the constants in the parametric equations to find the coordinates of a point which lies on the line. We can see the point with coordinates two, negative three must lie on this line. This is equivalent to substituting 𝑘 is equal to zero into the parametric equations. In exactly the same way, we can determine a vector parallel to the line by looking at the coefficients of the parameter 𝑘.
In the first equation, the coefficient of 𝑘 is five. So the horizontal component of our vector will be five. In the second equation, however, we can see that there is no 𝑘-term. So the coefficient of 𝑘 must be equal to zero, which means the vertical component of this vector is zero. Therefore, the vector five, zero must run parallel to the line. And it’s worth noting any nonzero scalar multiple of this vector will also be parallel to the line. For example, if we call our scalar 𝑟, we can set 𝑟 equal to negative one to note that the vector negative five, zero is also parallel to the line. It’s another choice for the direction vector. Similarly, we can choose our value of 𝑟 to be one-fifth. This would give us the vector one, zero, another possible direction vector for the line.
All of these are valid direction vectors of the line and can be considered the correct answer. However, we will just choose the vector which we can read off from the parametric equations. We’ll say that the direction vector of this line is the vector five, zero.
