Eroxl's Notes
Web Work 1 (MATH 317)

Problem 1

Below is a diagram of the dotted cube with edges of length 5. Suppose vectors , , and . Find the following vectors.

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(a).

(b).

(c).

(d).

Problem 2

Let be the vector and be the point . What is the endpoint of if it starts at ?

Problem 3

Which of the following Vectors Are Parallel?

(a). Please Select the Two Vectors that Are Parallel

  • [ ] A.
  • [ ] B.
  • [ ] C.
  • [x] D.
  • [ ] E.
  • [x] F.

(b). Please Select the Two Vectors that Are Parallel

  • [x] A.
  • [ ] B.
  • [ ] C.
  • [ ] D.
  • [ ] E.
  • [x] F.

Problem 4

The figure below shows two named points and in .

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(a). Part 1

(i). Find the Coordinate Vector for

(ii). Write the Vector as Two Nonzero, Nonparallel Vectors

(iii). Write the Vector as Three Nonzero, None of Which Are Parallel to the Others

(b). Part 2

(i). Write the Vector as Two Vectors, One Parallel to and the other Parallel to

(ii). Write the Vector as Two Vectors, One Parallel to and the other Parallel to

Problem 5

(a). Write the Vector in terms of the other Vectors

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(b). Write the Vector in terms of the other Vectors

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(c). Write the Vector in terms of the other Vectors

Pasted image 20260109151216.png

Problem 6

Find a vector that has the same direction as but has a length of 5.

Problem 7

Find the angle between the vectors and .

Problem 8

Find the scalar and vector projections of onto where and .

Problem 9

For what values of are the vectors and orthogonal.

Problem 10

Find the area of the parallelogram with vertices , , and

Problem 11

Suppose that the line is represented by and the plane is represented by

(a). Find the Intersection of the line and the Plane Write Your Answer as a point where , , and Are Numbers

(b). Find the Cosine of the Angle between the line and the Normal Vector of the Plane

Problem 12

Find the cross product

Problem 13

Find the vector and parametric equations for the line through the point and the point .

Problem 14

Find the vector and parametric equations for the line through the point and parallel to the vector .

Problem 15

Find the vector equation for the line of intersection of the planes and .

Problem 16

Find parametric equations for the arc of a circle of radius 4 from to .

Problem 17

Parameterize the line through and so that the points and correspond to the parameter values and respectively.

Problem 18

Find a vector function that represents the curve of intersection of the paraboloid and the cylinder . Use the variable for the parameter.

Problem 19

Find a vector parametrization of the ellipse centred at the origin in the -plane that has major diameter 16 along the -axis, minor diameter 12 along the -axis, and is oriented counter-clockwise. Your parametrization should make the point correspond to . Use as the parameter in your answer.

Problem 20

(a). Find a Parameterization for the line through the point and in the Direction of

(b). Find Conditions on so that the line You Found in part (a) Goes through the Origin. (Be Sure You Can Give a Reason for Your answer.) then Use Your Work to Give Two Distinct Triples that Result in the line Passing through the Origin

Any multiple (positive or negative) of the point will give a line that crosses through the origin as there is always a real number such that . and thus .

Problem 21

Find a parametrization of the circle of radius 3 in the -plane, centred at , oriented counterclockwise. The point should correspond to . Use as the parameter for all of your answers.

Problem 22

Consider an object following the path of the two-dimensional vector-valued function .

(a). When Does it Pass through the point ? if it Passes through that Point, Give the Value. if it Does not Pass through that Point, Enter None as the Answer

It does not pass through the point

(b). When Does it Pass through the point ? if it Passes through that Point, Give the Value. if it Does not Pass through that Point, Enter None as the Answer

It passes the point at

(c). When Does it Pass through the point ? if it Passes through that Point, Give the Value. if it Does not Pass through that Point, Enter None as the Answer

It passes the point at

(d). When is it at Rest? if it is at Rest at Some Point, Give the Value. if it is Never at Rest Enter None for the Answer

The object is never at rest as both the x and y velocity are never 0 at the same time.

Problem 23

(a). Find a Vector-parametric Equation for the Shadow of a Circular Cylinder in the -plane

(a). Find a Vector-parametric Equation for the Intersection of a Circular Cylinder and the Plane

Problem 24

Find the point of intersection , of the lines and