- Find the coordinates of the corner point where the three planes intersect.
Answer: (3, 4, 2)
We solve the system of three linear equations:
1) x + z = 5
2) y + z = 6
3) x + y = 7
From equation (1): z = 5 - x
From equation (2): z = 6 - y
Setting these equal: 5 - x = 6 - y, which gives y = x + 1
Substituting into equation (3): x + (x + 1) = 7
2x + 1 = 7, so x = 3
Then y = 3 + 1 = 4
And z = 5 - 3 = 2
The intersection point is (3, 4, 2). Option A results from solving incorrectly. Option C swaps x and y values. Option D represents a common algebraic error in the substitution step.
- Which statement correctly describes the alignment of the satellites?
Answer: The satellites lie on the same straight line path.
To check for collinearity, we find the vectors between the points and see if they are parallel.
Vector AB = B - A = (3-1, 5-2, 7-3) = (2, 3, 4).
Vector BC = C - B = (7-3, 11-5, 15-7) = (4, 6, 8).
We can see that BC = 2 * AB. Since one vector is a scalar multiple of the other and they share a common point (B), the points A, B, and C are collinear. Therefore, the satellites lie on the same straight line path.
A is incorrect because |BC| = 2|AB|. The magnitude of AB is √29 and the magnitude of BC is √116 = 2√29. C is incorrect as the vectors are parallel, not perpendicular.
- In this model, what is the compass direction of steepest ascent on the hillside?
Answer: A direction mostly East and slightly North.
The gradient of a plane f(x,y,z)=c is given by the normal vector. The direction of steepest ascent on the surface z = f(x,y) = 100 - 5x - 2y is given by the vector (-∂z/∂x, -∂z/∂y), which corresponds to the projection of the normal vector onto the xy-plane.
The normal vector to the plane 5x + 2y + z = 100 is n = (5, 2, 1). The projection of this vector onto the xy-plane is (5, 2, 0) or simply (5, 2). This vector represents the direction of the greatest rate of change of height, i.e., the direction of steepest ascent.
A vector (5, 2) in the xy-plane corresponds to moving 5 units in the positive x-direction (East) and 2 units in the positive y-direction (North). This is a direction that is mostly East and slightly North.
- What is the magnitude of the component of the force that acts along the track?
Answer: 3.46 N
The component of force F along direction d has magnitude given by the scalar projection formula: |(F ⋅ d)|/|d|
Calculate the dot product: F ⋅ d = (5)(1) + (2)(1) + (-1)(1) = 5 + 2 - 1 = 6
Calculate |d| = √(1² + 1² + 1²) = √3 ≈ 1.732
Component magnitude = 6/√3 = 6/1.732 ≈ 3.46 N
Option A gives |d| only. Option C gives |F| = √(25+4+1) = √30 ≈ 5.48. Option D gives the dot product value before division.
