4. Write a vector equation of the line through the point (5, -2, 3) and parallel to the vector v=[4, -3, 1]
To write a vector equation of a line through a given point and parallel to a given vector, we can use the parametric form of a vector equation.
Let's call the given point (x₁, y₁, z₁) and the given vector (a, b, c).
The vector equation of the line can be written as:
r = (x₁, y₁, z₁) + t(a, b, c)
where r is a position vector of any point on the line and t is a scalar parameter.
In this case, the given point is (5, -2, 3) and the given vector is [4, -3, 1].
Therefore, the vector equation of the line parallel to the vector v=[4, -3, 1] and passing through the point (5, -2, 3) is:
r = (5, -2, 3) + t(4, -3, 1)
So, the vector equation is r = (5 + 4t, -2 - 3t, 3 + t), where t is a scalar parameter that can take any real value.