Video: Finding the Position Vector of a Given Point on this Line Corresponding to a Given Parameter

Using point (4, 9, −9) and direction vector 〈−5, 1, −6〉, give the position vector 𝑟 of the point on this line corresponding to parameter 𝑡 = 9.

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Video Transcript

Using point four, nine, negative nine and direction vector negative five, one, negative six, give the position vector 𝑟 of the point on this line corresponding to parameter 𝑡 equals nine.

Let’s begin by identifying exactly what this question is asking us. We’re being asked to find a position vector 𝑟 of a point on a line defined by the point four, nine, negative nine and a direction vector negative five, one, negative six. Let’s begin by recalling the vector equation of a line. It’s 𝑟 equals 𝑟 naught plus 𝑡𝑑. In this equation, 𝑟 naught describes the vector that takes us from the origin to a point on this line. 𝑑 is the direction vector. And this is a little bit like the gradient or the slope in the formula for the Cartesian equation of a straight line. 𝑡 is just a real constant. So let’s begin by substituting what we know about our line into this formula.

The vector that gets us from the origin to a point on the line is given by four, nine, negative nine and the direction vector is stated in the question as negative five, one, negative six. And this means that the vector equation of our line is four, nine, negative nine plus 𝑡 times negative five, one, negative six. We’re looking to find the position vector of the point on a line that corresponds to the parameter 𝑡 equals nine. And so we substitute nine into our equation. And we obtain 𝑟 to be equal to four, nine, negative nine plus nine times negative five, one, negative six.

We can multiply this constant by the second vector by multiplying nine by each element; that’s by negative five, one, and negative six. And when we do, we obtain that the position vector 𝑟 is given by four, nine, and negative nine plus negative 45, nine, negative 54. We’re simply going to add each element in our vector. Four plus negative 45 is negative 41. Nine plus nine is 18. And negative nine plus negative 54 is negative 63. And we see that the position vector 𝑟 of the point on our line corresponding to parameter 𝑡 equals nine is negative 41, 18, negative 63.

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