The line r=(2,3,4)+µ(-1,0,3) and the point (6,0,2) form which plane: (a) r.(-1,0,3)=10 (b) r.(9,10,3)=60 (c) r.(2,3,4)=29 (d) r=(2,3,4)+µ(-1,0,3)+ù(6,0,2)
Please help!!! Thanks!
The required plane can be found by:
1. find two distinct points A, B on the line r.
2. Form two vectors from the point P to the two points on the line. These two vectors will lie in the required plane.
3. find the normal n to the plane by the cross product of the two vectors PA and PB.
4. Determine the plane from the normal n and any point passing through the plane, i.e. P, A or B.
Here are the details:
r : (2,3,4)+μ(-1,0,3)
Set μ to 0 and 1 to get two distinct points A & B.
A(2,3,4)
B(2-1, 3-0, 4+3)=B(1,3,7)
The point P(6,0,2) is in the plane.
Now calculate vectors PA and PB:
PA(2-6,3-0,4-2)=PA(-4,3,2)
PB(-5,3,5)
The normal n to the plane is given by the cross product of PA and PB, namely
n=PA X PB
=(9,10,3)
The plane passing through the line r and the point P is therefore of the form
9x+10y+3z+k=0 ......(1)
Substitute the coordinates of P into the above equation (1) to find k=-60.
The plane is flying at an altitude of 8 miles and is 175 miles from the runway, as measured from the ground. What angle will the plane's path make with the runway? Round your answer to the nearest hundredth.
To determine which plane the line r = (2, 3, 4) + µ(-1, 0, 3) and the point (6, 0, 2) form, we need to check which equation of the plane is satisfied by these points.
(a) r.(-1, 0, 3) = 10: This equation is a dot product of the vector (-1, 0, 3) with the position vector r of the line. It does not involve the point (6, 0, 2). Hence, this equation does not represent the plane formed by the line and the point.
(b) r.(9, 10, 3) = 60: Similarly, this equation involves the dot product of the vector (9, 10, 3) with the position vector r. It does not take into account the given point (6, 0, 2). Therefore, this equation does not represent the plane either.
(c) r.(2, 3, 4) = 29: This equation involves the dot product of the vector (2, 3, 4) with the position vector r. Since this equation involves the vector representation of the line and not just the point (6, 0, 2), we can check if this equation is satisfied to determine the plane.
(d) r = (2, 3, 4) + µ(-1, 0, 3) + ù(6, 0, 2): This equation explicitly represents the line in vector form along with the point (6, 0, 2). However, it does not represent a plane equation.
Therefore, the plane formed by the line r = (2, 3, 4) + µ(-1, 0, 3) and the point (6, 0, 2) is given by the equation r.(2, 3, 4) = 29.