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Determine the vector form of the equation of the straight line passing through the point negative one, negative five, four and parallel to the vector negative three, five, one.
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Alright, so here we have a point negative one, negative five, four and a vector with components negative three, five, one.
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And we imagine a line that passes through this point and is parallel to the given vector.
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Here, we want to solve for the vector form of the line’s equation.
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To start figuring this out, we can recall how we generally write the vector form of a line.
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It involves two vectors, one that goes from the origin of a coordinate frame to a known point on the line, we’ll say it has coordinates 𝑥 one, 𝑦 one, and 𝑧 one, and a second vector that is parallel to the line’s axis.
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So then a point on the line and a vector parallel to the line are the two things we need to write the equation of a line in vector form.
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And we see that in this scenario, that’s exactly what we’re given.
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We can write then that 𝐫, the vector representing our line, equals a vector from the origin of our coordinate frame to our known point, negative one, negative five, four, plus a scale factor we’ll call 𝑡 multiplied by our vector that’s parallel to the line.
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This is the equation of our line in vector form.