Vectors Questions and Answers

Example: The diagram shows a cuboid whose vertices are Q. A, B, C, D, E, F and G. Vectors a, b and c are the position vectors of the vertices A, B and C respectively. The points and w and x the points Y and Z lie on OA and EF such that OW: WX :XA = 1:3:1=EY: YZ: ZF. Prove that the diagonals WY and XZ bisect each other.

To be able to: Use vectors to solve generalised geometric problems Vectors are often used to prove general properties about geometric arrangements. In the following examples, no coordinates or specific vectors will be given. We often make use of the following fact: In n dimensions, each vector is a unique linear combination of a non-parallel vectors. Example: OABC is a parallelogram. The vectors ar and e are the position vectors of points 4 and C respectively. Prove that the diagonals of OABC bisect each other. Method 1 Let point P be the intersection of the diagonals AC and OB Method 2 Let points P and Q be the mid-points of AC and OB respectively.

Let L₁ be the line passing through the point P₁=(-1, 0, 6) with direction vector d₁[-1-3,-1, 1]ᵗ, and let L₂ be the line passing through the point P₂=(7, 3, 11) with direction vector d₂[-3, -5.-3] ᵗ. Find the shortest distance d between these two lines, and find a point Q₁ on I₁., and a point Q₂ on L₂ so that d(Q₁,Q2)=d. Use the square root symbol 'v' where needed to give an exact value for your answer. d = 0 Q1= Q2=

A class consists of 60 % men and 40% women. Blond men compose 25% of the class, and blond women make up 20% of the class. If a student is chosen at random and is found to be a male, what is the probability that the student is blond? A 0.34 B. 0.42 C. 0.54 D. 0.63

Suki is making lemonade to bring to the beach. She has two containers. One holds one gallon and the other holds 6 quarts. If she fills both containers, how many cups of lemonade will she have? She will have ______ cups of lemonade.

A bicycle is traveling at 21 miles per hour. How many feet will it cover in 45 seconds? Round your answer to the nearest tenth of a foot

(1) Is A similar to a diagonal matrix? If so, find a matrix B which diagonalize A. (2) Find the eigenvalues and bases for the eigenspaces of A³

In R4, consider the inner product (x, y) = x1y1 + 5x2y2 + x3y3 + 3x4y4. Let V be the nullspace of A = [1 0 1 -1 0 1 -1 2] a. The dimension of V is (Give a correct answer to see the next part of the question.)

Given the vectors (1, 0, 1) and (1, 2, -3). Write the steps necessary to show that these vectors form a basis for R³. If the vectors do not form a basis for R³, state the reason(s) why. Your steps should make use of spanning and linearly independence. From your analysis, give a clear and concise conclusion. Also as an addendum, include what defines a "linear algebra" and give two examples. Does your conclusion make sense? Please explain.

Every vector of P₂ is also a vector of P3. Select one: True False Let W be the set of vectors of the form A = a b of M2x2. The set W is a subspace of M2x2 b 1 Select one: True False

Vector u has initial point at (8, 6) and terminal point at (-6, 12). Which are the magnitude and direction of u?

The set S = {V1, V2, V3} where v₁ = (-1, 1, 1), v₂ = (1, -1, 1), v3 = (1, 1, -1) is a basis for R³. the vector w whose coordinate vector relative to this basis is (w)s = (3,-2, 4) is w =(_____)

In each answer choice a point is given along with a glide reflection. Which of the following is correctly stated? Select the correct answer below: (2,7) glide reflected along V =< 0,2> and across the y-axis is (2,-9), The transformation of (2, 3) translated by < 1. 1 > and then reflected in the vaxis is a valid glide reflection. (2,3) glide reflected along V=<1,0> and then reflected across the x-axis gives (3,-3). (1,4) glide reflected along V=< 3,3> and y = x gives (4,7).

Given the vector v =-3/√3,1; find the direction angle of this vector. a) 5л6 b) 2π3 c) -π3 d) л6 e) 0 f) None of the above.

Let A=(2,2,3), B=(5,-1,4) and C=(1,7,5). a) Find whether vectors A, B and C are linearly dependent/independent. b) Find ker(M) where columns of matrix M are vectors A, B and C.

The initial and terminal points of vector v are (2, -6) and (8, 1), respectively. (a) Write v in component form. (b) Write v as the linear combination of the standard unit vectors i and j.

5) Find the equation of the hyperbola based on the given information: A) Vertices V(2, -3) and V'(2, 7); one focus at F(2,-5) B) Foci at F(-2, 3) and F'(-2,9); one vertex at V(-2,4)

The planes 3x + 4y + 4z = -25 and 2z - (4x + 5y) = 3 are not parallel, so they must intersect along a line that is common to both of them. The vector parametric equation for this line is L(t) =

Given vectors a = (5, 3, 1) and b = (-1,3,-2), calculate the following: 1.3a-b II. a b III. a x b IV. proja b

A rectangular field is 0.35 kilometers long and 0.2 kilometers wide. What is the area of the field in square meters? Do not round your answer. Be sure to include the correct unit in your answer.

Over a two hour time period a snail moved 46 inches. How far is this in yards? Write your answer as a whole number or a mixed number in simplest form. Include the correct unit in your answer.

Subtract. Write your answer as a mixed number in simplest form.

The price of a condominium is $132,000. The bank requires a 5% down payment and one point at the time of closing. The cost of the condominium is financed with a 30-year fixed-rate mortgage at 8.5%. Use the following formula to determine the regular payment amount. Complete parts (a) through (e) below.

Let à = (-3, 4) and b = (5,-2). Find the projection of b onto a. proja b =

Find an equation of the plane. the plane through the point (1, 8, 3) and with normal vector 4i + 3j + 5k

Add the given vectors by components. A=380, θA=220.9° B=233, θB=294.2°

Find the coordinate vector of w relative to the basis S = (u₁, u₂) for R2. Let u₁=(4,-5), u₂ = (5, 7), w = (1,1). (w)s=

A truck with 26-in.-diameter wheels is traveling at 50 mi/h. Find the angular speed of the wheels in rad/min, "hint convert miles to inches & hours to minutes: rad/min How many revolutions per minute do the wheels make? rpm