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Example Video Example Describe the curve defined by the vector function r(t) = ⟨2+t, 3+4t, −5+2t⟩ Solution The corresponding parametric equations are x = , y = 3+4t, z = which we recognize from the equations x = x0+at, y = y0+bt, z = z0+ct as parametric equations of a line passing through the point (2, 3, −5) and parallel to the vector ⟨1, 4, 2⟩. Alternatively, we could observe that the function can be written as r = r0+tv, where r0 = ⟨2, 3, −5⟩ and v = , and this is the vector equation of a line as given by the equation r = r0+tv

Example Video Example Describe the curve defined by the vector function r(t) = ⟨2+t, 3+4t, −5+2t⟩ Solution The corresponding parametric equations are x = , y = 3+4t, z = which we recognize from the equations x = x0+at, y = y0+bt, z = z0+ct as parametric equations of a line passing through the point (2, 3, −5) and parallel to the vector ⟨1, 4, 2⟩. Alternatively, we could observe that the function can be written as r = r0+tv, where r0 = ⟨2, 3, −5⟩ and v = , and this is the vector equation of a line as given by the equation r = r0+tv

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Example Video Example Describe the curve defined by the vector function
r ( t ) = ⟨ 2 + t , 3 + 4 t , − 5 + 2 t ⟩
Solution The corresponding parametric equations are
x = ◻ , y = 3 + 4 t , z = ◻
which we recognize from the equations
x = x 0 + a t , y = y 0 + b t , z = z 0 + c t
as parametric equations of a line passing through the point ( 2 , 3 , − 5 ) and parallel to the vector ⟨ 1 , 4 , 2 ⟩ . Alternatively, we could observe that the function can be written as r = r 0 + t v , where r 0 = ⟨ 2 , 3 , − 5 ⟩ and v = ◻ , and this is the vector equation of a line as given by the equation r = r 0 + t v

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