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Give the vector equation of the line through the point three, seven, negative seven with direction vector zero, negative five, seven.
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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 𝑡𝐯, 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.
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𝑡 is a scalar multiple.
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If we look at the information that we’re given in the question, we can see that we have a direction vector.
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And we’ve got a point on the line which can be written as a position vector.
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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.
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We can then simply plug in these two vectors into the vector form.
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𝐫 equals the position vector three, seven, negative seven plus 𝑡 times the direction vector zero, negative five, seven.
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And so that’s the answer for the vector equation of the line.