MATH Vector Calc Exam 1

Q3 Use vectors to determine if the following planes are perpendicular. 2(x−5) + 6(y+1) − 8(z−3) = 0 −3(x−1) + 8y + 2z = 0

A normal vector to the first plane is 2i+6j-8k. A normal vector to the second plane is -3i+8j+2k. The dot product of these vectors is (2)(-3)+(6)(8)+(-8)(2)=26. If the planes were perpendicular, their normal vectors would be perpendicular and the dot product would be zero

Q2) Some airplanes can create a sonic boom. A sonic boom carpet is a region on the ground where the sonic boom is heard directly from the airplane. The width of the carpet, W, can be expressed as a function of a, the air temperature on the ground, and v, the vertical temperature gradient at the airplane's altitude. Suppose W(a,v) = k (sqrt(a/v)) where k is a positive constant. (a) If v is fixed, is W an increasing or decreasing function of a? (b) If a is fixed, is W an increasing or decreasing function of v?

a) Increasing B) Decreasing

Q1) (a) Find u · v, if we know that || u || = 16 and u makes an angle of 𝜋/3 with v = 3 i + 2 j. b) The velocity of a plane is given by p = 100(3 i + 2 j ). What is the tangent of the angle that the plane's heading makes from due east? Assume no wind is affecting the plane. tan(𝜃) =

V= ( ||v|| cos(θ) ) + ( ||v|| sin(θ) )

Q2 (a) Which point is on the xz-plane? (4, 9, 0) (4, 0, 9) (0, 4, 9) Which point is on the x-axis? (0, −4, 0) (0, 0, −4) (−4, 0, 0) Which point is closest to the yz-plane? (10, 6, 4) (4, 10, 6) (b) Find the shortest distance from (4, 9, 10) to the y-axis. Give an exact answer.

(a) A point on the xz-plane must have a y-coordinate equal to 0. (4, 0, 9) A point on the x-axis must have y and z-coordinates equal to 0. (−4, 0, 0) The point closest to the yz-plane must have an x-coordinate closest to 0. (4, 10, 6) (b) closest point on the y axis is (0,9,0) use the distance formula sqrt((x1-x2)^2 - (y1-y2)^2 - (z1-z2)) sqrt( 4^2 + 0^2 +10^2_ sqrt(116)

Q2 In each case, determine if the quantity or expression is a scalar, vector, or neither. Assume all vectors are non-zero vectors and all constants are non-zero constants. (a) c ( u · v ) (b) (u + v)/w (c) The output of a function f(x, y) at a point. (d) (a × b)/(a · b) (e) ( u × v ) × w

(a) Dot product is a scalar. Here we are only multiplying that scalar by a constant. (b) Dividing by a vector is not defined in real numbers. (c) A function value is a number. (d) The numerator is a vector. The denominator is a scalar. (e) The object in the parentheses is a vector. When we take the cross product of this vector with another vector, we get a vector.

Q2) True or false? Assume the vectors are any non-zero vectors in 3-space. a) a × b = b × a b) u · ( u × w ) = 0 c) a × b = || a || || b || sin(𝜃)

(a) Order matters when taking cross product. When we change the order, our resulting vector is in the opposite direction.(b) u × w is a vector that is perpendicular to both u and w .So when we take the dot product of such a vector with u , our dot product is 0.(c) The left hand side is a vector. The right hand side is a scalar.If you thought this was true because you remembered the formula on the right hand side, that formula is the magnitude of the cross product, not the cross product itself.

Q2 (a) Write an equation for a standard paraboloid with vertex (0, 0, 14) and opening up. (b) Write an equation for a plane parallel to the xz-plane and passing through the point (4, 9, −14). (c) What surface is represented by 81 = y^2 + z^2? (d) Which of the following equations represents the top half of a sphere of radius 10 centered at (−3, 0, 0)?

(a) Start with a regular paraboloid and shift up. (b) A plane parallel to the xz plane has the form y = c. (c) Because x does not appear and cross sections are all circles of the same size, this is a horizontal cylinder with x axis. (d) Start with a regular sphere and solve for z. Choose positive square root.

Q2) Use the following vectors to answer each part. Give exact simplified answers. u = 3 i + 5k and w = −4 i + j − 2k (a) Find a non-zero vector in the xz-plane that is perpendicular to u, but not w. (b) Find a non-zero vector that is perpendicular to both u and w (c) Find the exact area of the parallelogram formed by u and w

(a) There are many answers, but the vector cannot have a j component because it is in the xz-plane and the dot product with u must be zero.(c) A property of cross product is that its magnitude is the area of the parallelogram. The answer here is the magnitude of the vector answer in part (b).

Q3 The figure below is a contour diagram of the monthly payment on a loan, as a function of the interest rate, R, and the amount of money borrowed, (a) Estimate ≈ $ 100 (b) Give a good practical interpretation of your answer in part (a). Use a complete sentence(s) and everyday language. Write in a way that a person who is not in a math class would understand

An example of a practical interpretation: "If you borrow 5500 dollars with a loan that charges 4 percent interest, then your monthly loan payments will be about 100 dollars". This problem with practical interpretation was illustrated in the Loan Example video https://www.showme.com/sh?h=jepydzE in