Give a vector parametric equation for the line through the point (2,4,−5)(2,4,−5) that is parallel to the line ⟨−5−5𝑡,2+𝑡,5𝑡⟩

The vector parametric equation for a line in 3D space is given by the formula:

r(t) = r_0 + t * v

where r_0 is a position vector for a point on the line, v is a vector parallel to the line, and t is a parameter.

Given the point (2,4,-5), we can write the position vector r_0 as ⟨2,4,-5⟩.

The line ⟨-5-t,2+t,5t⟩ is parallel to the vector v = ⟨-1,1,5⟩ (we get this by taking the coefficients of t in the given line).

So, the vector parametric equation for the line through the point (2,4,-5) that is parallel to the line ⟨-5-t,2+t,5t⟩ is:

r(t) = ⟨2,4,-5⟩ + t * ⟨-1,1,5⟩ = ⟨2-t, 4+t, -5+5t⟩

So, the line is parametrized by ⟨2-t, 4+t, -5+5t⟩.