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chandlerstandi8294
12.02.2021 •
Mathematics
1. Find symmetric equations for the line that passes through the point
(4, −4, 8) and is parallel to the vector −1, 4, −3
(b) Find the points in which the required line in part (a) intersects the coordinate planes.
point of intersection with xy-plane
point of intersection with yz-plane
point of intersection with xz-plane
2. Find an equation for the plane consisting of all points that are equidistant from the points
(−6, 4, 1) and (2, 6, 5).
3. Find an equation of the plane.
The plane that passes through the point (−2, 1, 1) and contains the line of intersection of the planes
x + y − z = 3 and 4x − y + 5z = 5
Solved
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Ответ:
The answer is below
Step-by-step explanation:
1
a) The symmetric equations for the line that passes through the point (a, b, c) and is parallel to the vector (e, f, g) is:
Therefore using the above equation to Find symmetric equations for the line that passes through the point (4, −4, 8) and is parallel to the vector (−1, 4, −3) we get:
b)i) The line would intersect the xy plane where z = 0. Hence:
ii) The line would intersect the yz plane where x = 0. Hence:
iii) The line would intersect the xz plane where y = 0. Hence:
2)
Let the points be A(−6, 4, 1) and B(2, 6, 5). Let O be the midpoint of the two points A and B. Therefore O is the average of the x coordinates, y coordinates and z coordinate.
The normal vector (n) in the direction of line between A and B is:
n = AB = B - A = (2, 6, 5) - (−6, 4, 1) = (8, 2, 4)
n = 8x + 2y + 4z
The equation of the plane based on the normal vector and the midpoint 0 is:
Plane = 8x + 2y + 4z = 8(-2) + 2(5) + 4(3) = 6
Therefore:
8x + 2y + 4z = 6
4x + y + 2z = 3
3) The normal vectors to the plane are:
The point of intersection O of the two planes is normal to the normal vectors, hence:
A point that lies on both plane is gotten by substituting z = 0, hence:
x + y - (0) = 3, and 4x - y + 5(0) = 5
x + y = 3 and 4x - y = 5
Solving simultaneously gives x = 8/5, y=7/5
From this two points we get:
AB = (-2-8/5, 1 - 7/5, 1-0) = (-18/5, -2/5, 1)
The vector normal to the plane (n) = (4, -9, -5) * (-18/5, -2/5, 1) = (-11, 14, -34)
Ответ:
I only see one raffle, but the expected value of one ticket is $2. (I think)
Step-by-step explanation:
Have a nice day!