Which of the following is the vector equation of a plane that contains the line mc031-1.jpg and the point mc031-2.jpg?
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To find the vector equation of a plane that contains a line and a point, we first need to find two vectors that lie in the plane.
Let's start by finding the first vector. We can do this by subtracting the coordinates of two points on the line. Let's choose two points on the given line:
Point 1: (1, -2, 3)
Point 2: (-1, 4, 2)
The vector between these two points is given by:
v1 = Point 2 - Point 1
v1 = (-1, 4, 2) - (1, -2, 3)
v1 = (-2, 6, -1)
Now, let's find the second vector. This vector can be found by subtracting the coordinates of a point on the line from the coordinates of the given point outside the line.
The given point outside the line is (2, 0, -4).
v2 = (Given point) - (Point on the line)
v2 = (2, 0, -4) - (1, -2, 3)
v2 = (1, 2, -7)
Now, we have two vectors that lie in the plane. We can use these vectors and the given point to write the vector equation of the plane.
P0 = (2, 0, -4) (Given point)
The vector equation of the plane is given by:
r = P0 + sv1 + tv2
where r is any position vector on the plane, s and t are scalar parameters, P0 is the given point, v1 is the first vector, and v2 is the second vector.
Therefore, the vector equation of the plane that contains the given line and point is:
r = (2, 0, -4) + s(-2, 6, -1) + t(1, 2, -7)
To determine the vector equation of a plane that contains both a given line and a point, we need to find two non-parallel vectors that lie on the plane.
Given the line mc031-1.jpg, we can find a direction vector for this line by subtracting the coordinates of any two points on the line. Let's choose points P₁(mc031-3.jpg) and P₂(mc031-4.jpg) on the line. The direction vector, mc031-5.jpg, is given by:
mc031-6.jpg
Next, let's find a vector that connects the point mc031-2.jpg, which lies on the plane, to a point on the line. We can use the position vector from the origin to the point mc031-2.jpg, defined as mc031-7.jpg.
To obtain the second vector, we subtract the position vectors of any two points that lie on the line. Let's use points P₃(mc031-8.jpg) and P₄(mc031-9.jpg) on the line. The vector connecting point mc031-2.jpg to the line, mc031-10.jpg, is given by:
mc031-11.jpg
Now, we can write the vector equation of the plane using the point-normal form. The point-normal form equation of a plane is:
mc031-12.jpg
where mc031-7.jpg is the position vector of the given point, and mc031-5.jpg and mc031-10.jpg are the two non-parallel vectors lying on the plane.
Substituting the values we found earlier, the vector equation of the plane is:
mc031-13.jpg