Topic 4 Three-Dimensional Trigonometry

Question 1

What is the definition of three-dimensional trigonometry?

Answer 1

Three-dimensional trigonometry is the study of the relationships between the angles and sides of three-dimensional figures.

Question 2

What are the three axes in a three-dimensional coordinate system?

Answer 2

The x-axis, y-axis, and z-axis.

Question 3

Fill in the blank: The distance between two points in 3D space can be calculated using the ______ formula.

Answer 3

distance

Question 4

What is the distance formula in three dimensions?

Answer 4

d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)

Question 5

What is the formula for the volume of a rectangular prism?

Answer 5

Volume = length × width × height

Question 6

True or False: The sum of the angles in any triangle is always 180 degrees.

Answer 6

True

Question 7

What is the relationship between the sides and angles in a right triangle?

Answer 7

The sine, cosine, and tangent functions relate the angles to the lengths of the sides.

Question 8

What is the cosine rule for a triangle in three dimensions?

Answer 8

c² = a² + b² - 2ab cos(C)

Question 9

Fill in the blank: The ______ of a vector is the length of the vector.

Answer 9

magnitude

Question 10

How do you calculate the magnitude of a vector (x, y, z)?

Answer 10

Magnitude = √(x² + y² + z²)

Question 11

What is the sine rule for a triangle in three dimensions?

Answer 11

a/sin(A) = b/sin(B) = c/sin(C)

Question 12

What is the projection of a vector onto another vector?

Answer 12

The projection is a vector that represents the component of one vector in the direction of another.

Question 13

True or False: The dot product of two vectors is a scalar.

Answer 13

True

Question 14

What is the dot product formula for vectors A and B?

Answer 14

A · B = |A| |B| cos(θ)

Question 15

What does the cross product of two vectors produce?

Answer 15

A vector that is perpendicular to both of the original vectors.

Question 16

Fill in the blank: The formula for the cross product of vectors A and B is ______.

Answer 16

A × B = |A||B|sin(θ)n

Question 17

What is the angle between two vectors A and B if A · B = 0?

Answer 17

90 degrees (the vectors are perpendicular).

Question 18

What is the formula for the area of a triangle defined by three points in 3D space?

Answer 18

Area = 0.5 |AB × AC|

Question 19

True or False: The coordinates of a point in 3D space can be expressed as (x, y, z).

Answer 19

True

Question 20

What is the equation of a plane in three-dimensional space?

Answer 20

Ax + By + Cz + D = 0

Question 21

What does the normal vector of a plane represent?

Answer 21

A vector that is perpendicular to the plane.

Question 22

Fill in the blank: The ______ of a line in 3D can be defined using parametric equations.

Answer 22

equation

Question 23

What are the parametric equations of a line in 3D?

Answer 23

x = x0 + at, y = y0 + bt, z = z0 + ct

Question 24

What is the relationship between the lengths of the sides and the angles in a right triangle?

Answer 24

The lengths of the sides relate to the angles via the sine, cosine, and tangent functions.

Question 25

True or False: The coordinates of a vector are always positive.

Answer 25

False

Question 26

What is the formula to find the angle between two vectors A and B?

Answer 26

θ = cos⁻¹((A · B) / (|A||B|))

Question 27

Fill in the blank: The ______ of a vector can be found by dividing the vector by its magnitude.

Answer 27

unit vector

Question 28

What is the formula for the unit vector of vector A?

Answer 28

 = A / |A|

Question 29

What does it mean for two vectors to be orthogonal?

Answer 29

They are perpendicular to each other.

Question 30

Fill in the blank: The ______ of a triangle can be calculated using the cross product of two of its sides.

Answer 30

area

Question 31

What is a scalar triple product?

Answer 31

A · (B × C), giving the volume of the parallelepiped formed by the vectors.

Question 32

True or False: The volume of a tetrahedron can be calculated using the scalar triple product.

Answer 32

True

Question 33

What is the formula for the volume of a tetrahedron given by vectors A, B, and C?

Answer 33

Volume = (1/6) |A · (B × C)|

Question 34

What is the significance of the angle θ in the context of the cosine rule?

Answer 34

It is the angle opposite the side whose length is being calculated.

Question 35

Fill in the blank: The ______ of two vectors can be used to find the area of the parallelogram formed by them.

Answer 35

cross product

Question 36

What does a non-zero scalar triple product indicate?

Answer 36

The vectors are not coplanar.

Question 37

What is the relationship between the sine and cosine of an angle?

Answer 37

sin²(θ) + cos²(θ) = 1