Question Video: Finding the Vector Equation of the Line through a Given Point with a Given Direction Vector Mathematics

Give the vector equation of the line through the point (3, 7, βˆ’7) with direction vector (0, βˆ’5, 7).

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Video Transcript

Give the vector equation of the line through the point three, seven, negative seven with direction vector zero, negative five, seven.

We should remember that when we need to write an equation in vector form, it will be in the form 𝐫 equals 𝐫 sub zero plus 𝑑𝐯, where 𝐫 is the position vector of a general point on the line, 𝐫 sub zero is a position vector of a given point on the line, and 𝐯 is the direction vector. 𝑑 is a scalar multiple. If we look at the information that we’re given in the question, we can see that we have a direction vector. And we’ve got a point on the line which can be written as a position vector. As we navigate from the origin to the point three, seven, negative seven, then we can write this as the position vector three, seven, negative seven.

We can then simply plug in these two vectors into the vector form. 𝐫 equals the position vector three, seven, negative seven plus 𝑑 times the direction vector zero, negative five, seven. And so that’s the answer for the vector equation of the line.

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