Vectors and the Geometry of Space: True-False Quiz

in MES Science • 2 days ago (edited)

▶️ Watch on 3Speak - Odysee - BitChute - Rumble - YouTube - PDF notes

In this video I go over the end-of-chapter True-False Quiz as a review for the Vectors and the Geometry of Space chapter. There are 22 questions in total covering vectors, cross products, dot products, lines, surfaces, and much more. This is a great review of the videos I had covered earlier and helps crystalize the topics covered.

Timestamps

  • Intro: 0:00
  • Calculus Book Reference: 0:58
  • Sections in Calculus Book Chapter: 1:40
  • Topics to Cover: 2:50
  • True-False Quiz: 3:03
  • Question 1: 3:17
  • Question 2: 4:21
  • Question 3: 7:56
  • Question 4: 9:41
  • Question 5: 12:10
  • Question 6: 14:15
  • Question 7: 15:58
  • Question 8: 17:17
  • Question 9: 21:02
  • Question 10: 28:12
  • Question 11: 33:25
  • Question 12: 40:48
  • Question 13: 50:19
  • Question 14: 56:22
  • Question 15: 59:16
  • Question 16: 1:01:21
  • Question 17: 1:01:55
  • Question 18: 1:03:18
  • Question 19: 1:05:57
  • Question 20: 1:07:51
  • Question 21: 1:09:26
  • Question 22: 1:11:23
  • Outro: 1:13:51

Video sections - Vectors and the Geometry of Space

View Video Notes Below!

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Vectors and the Geometry of Space: True-False Quiz

Vectors Geometry Quiz.jpeg

Calculus Book Reference

Note that I mainly follow along the book:

  • Calculus: Early Transcendentals 7th Edition by James Stewart
  • Note: In some earlier videos I used the 6th edition.

For this video, I used this website for the Quiz solutions: Link

Sections in Calculus Book Chapter

I have made a list of the sections in this particular chapter with links to the Hive post of the videos I have already finished.

Note that I have started splitting each long video into sections and made a playlist for each.

Vectors and the Geometry of Space

  1. 3-D Coordinate Systems
  2. Vectors
  3. The Dot Product - Playlist
  4. The Cross Product - Playlist
  5. Equations of Lines and Planes - Playlist
  6. Cylinders and Quadric Surfaces - playlist
  7. Review - playlist
  8. Problems Plus - ▶️

This links are also on MES Links: https://mes.fm/links

Topics to Cover

Note that the timestamps will be included in the video description for each topic listed below.

  • Question 1
  • Question 2
  • Question 3
  • Question 4
  • Question 5
  • Question 6
  • Question 7
  • Question 8
  • Question 9
  • Question 10
  • Question 11
  • Question 12
  • Question 13
  • Question 14
  • Question 15
  • Question 16
  • Question 17
  • Question 18
  • Question 19
  • Question 20
  • Question 21
  • Question 22

True-False Quiz

Determine whether the statement is true or false.

If it is true, explain why.

If it is false, explain why or give an example that disproves the statement.

Question 1

If u = <u1, u2> and v = <v1, v2>, then u ∙ v = <u1v1, u2v2>.

Solution: False

This is false, the Dot Product is a scalar and NOT a vector.

image.png

Question 2

For any vectors u and v in V3, |u + v| = |u| + |v|.

Solution: False

Consider the following example:

image.png

Hence, the statement is false.

Question 3

For any vectors u and v in V3, |u ∙ v| = |u||v|.

Solution: False

This false, as the formula needs a cosine function.

The geometric formula for the dot product is shown below.

image.png

Thus, the magnitude of the dot product would be:

|u ∙ v| = |u||v||cos θ|

Question 4

For any vectors u and v in V3, |u x v| = |u||v|.

Solution: False

This is false, the length of the cross product formula has a sine function in it.

image.png

However, for perpendicular vectors, the formula would be true.

image.png

Question 5

For any vectors u and v in V3, u ∙ v = v ∙ u.

Solution: True

This is true, and is a property of the dot product.

image.png

We can see this is the case:

image.png

Question 6

For any vectors u and v in V3, u x v = v x u.

Solution: False

This is false because the direction of the cross product follows the right hand rule, in which the fingers curl in the direction of the angle from u to v, and the thumb gives the positive cross product value.

image.png

Reversing the order of the vectors in the cross product would result in the thumb pointing in the opposite direction, hence the cross product will be negative.

image.png

Question 7

For any vectors u and v in V3, |u x v| = |v x u|.

Solution: True

This is true, since reversing the vectors in the cross product just gives a negative value, but the magnitude is the same.

image.png

Question 8

For any vectors u and v in V3 and any scalar k, k(u ∙ v) = (ku) ∙ v.

Solution: True

This is true and is a basic arithmetic multiplication property of the dot product.

image.png

We can see this is the case:

image.png

Question 9

For any vectors u and v in V3 and any scalar k, k(u x v) = (ku) x v.

Solution: True

This is true and is a basic arithmetic multiplication property of the cross product.

image.png

We can see this is the case:

image.png

Question 10

For any vectors u, v, and w in V3, (u + v) x w = u x w + v x w.

Solution: True

This is true and is another property of the cross product.

image.png

We can see why this is the case:

image.png

Question 11

For any vectors u, v, and w in V3, u ∙ (v x w) = (u x v) ∙ w.

Solution: True

This is true as it is a property of the scalar triple product:

image.png

We can see this is the case:

image.png

Question 12

For any vectors u, v, and w in V3, u x (v x w) = (u x v) x w.

Solution: False

This expression u x (v x w) is known as the Vector Triple Product.

image.png

Let's see if the given statement is true for the following example:

image.png

Thus the statement is false.

Question 13

For any vectors u and v in V3, (u x v) ∙ u = 0.

Solution: True

This is true because the cross product gives a vector that is perpendicular to both vectors, and the dot product between the cross product and one of its vectors is equal to zero if they are perpendicular to each other, which they are.

image.png

image.png

We can see this is true by computing the scalar triple product:

image.png

image.png

Question 14

For any vectors u and v in V3, (u + v) x v = u x v.

Solution: True

This is true because of the distributive property of the cross product (Question 10), and that the cross product of a vector with itself is equal to zero.

image.png

image.png

Question 15

The vector <3, -1, 2> is parallel to the plane 6x - 2y + 4z = 1.

Solution: False

The normal vector to the plane can be found in the coefficients of the linear equation of a plane.

image.png

The normal vector is <6, -2, 4>, and our given vector is just the normal vector divided by 2.

image.png

Thus the given vector is not parallel to the plane because it is parallel to the normal vector to the plane, that is, it is perpendicular to the plane.

Thus the statement is false.

Question 16

A linear equation Ax + By + Cz + D = 0 represents a line in space.

Solution: False

This is false, the linear equation represents a plane and not a line.

Question 17

The set of points {(x, y, z) | x2 + y2 = 1} is a circle.

Solution: False

This is false, the equation represents a circular cylinder.

image.png

We can also graph the cylinder in GeoGebra: https://www.geogebra.org/calculator/vgjh47g3

image.png

image.png

Question 18

In R3 the graph of y = x2 is a paraboloid.

Solution: False

This is false, the graph y = x2 in 3 dimensions represents a parabolic cylinder.

image.png

Graphing in GeoGebra, we can see that the equation is a horizontal parabolic cylinder: https://www.geogebra.org/calculator/vgjh47g3

image.png

image.png

Question 19

If u ∙ v = 0, then u = 0 or v = 0.

Solution: False

This is false because the dot product is zero if the vectors are perpendicular to each other regardless if one is equal to zero or not.

For example:

image.png

Question 20

If u x v = 0, then u = 0 or v = 0.

Solution: False

This is false because the cross product is equal to zero if the vectors are parallel regardless if any is zero or not.

For example:

image.png

Question 21

If u ∙ v = 0 and u x v = 0, then u = 0 or v = 0.

Solution: True

This is true because the dot product being zero means both vectors are perpendicular while the cross product being zero means both vectors are parallel.

This is only the case if one of the vectors is zero.

image.png

Question 22

If u and v are in V3, then |u ∙ v| ≤ |u||v|.

Solution: True

This is true because the geometric formula for the dot product includes a cosine function, whose absolute value is at a maximum of 1.

image.png

image.png

This is also known as the Cauchy-Schwarz Inequality.

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