Vectors

Question 1

What are scalars?

Answer 1

Quantities specified by magnitude only.

Examples include temperature, time, and density.

Question 2

What are vectors?

Answer 2

Quantities requiring both magnitude and direction.

Examples include force, velocity, and electric field.

Question 3

How are vectors often represented?

Answer 3

In bold (e.g., a), with arrows (→a), or underlined (a).

Question 4

What is vector addition?

Answer 4

The sum of two vectors, denoted as 𝑐 = 𝑎 + 𝑏.

Question 5

What are the properties of vector addition?

Answer 5

• Commutative: 𝑎 + 𝑏 = 𝑏 + 𝑎
• Associative: 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐

Question 6

What is vector subtraction?

Answer 6

The difference of two vectors, denoted as 𝑎 − 𝑏 = 𝑎 + (−𝑏).

Question 7

What is a special case of vector subtraction?

Answer 7

𝑎 − 𝑎 = 0 (zero vector with no magnitude or direction).

Question 8

What happens when a vector is multiplied by a scalar?

Answer 8

Changes the magnitude by |𝜆| and keeps the direction the same unless 𝜆 < 0.

Question 9

What are the properties of multiplying a vector by a scalar?

Answer 9

• Commutative: (𝜆𝜇)𝑎 = 𝜆(𝜇𝑎) = 𝜇(𝜆𝑎)
• Distributive over vector addition: 𝜆(𝑎 + 𝑏) = 𝜆𝑎 + 𝜆𝑏
• Distributive over scalar addition: (𝜆 + 𝜇)𝑎 = 𝜆𝑎 + 𝜇𝑎

Question 10

What defines basis vectors?

Answer 10

Any three non-coplanar vectors (𝑒₁, 𝑒₂, 𝑒₃) in 3D can form a basis.

Question 11

How can a vector a be expressed in terms of basis vectors?

Answer 11

𝑎 = 𝑎₁𝑒₁ + 𝑎₂𝑒₂ + 𝑎₃𝑒₃.

Question 12

What are the unit vectors in Cartesian coordinates?

Answer 12

• 𝑖 = (1, 0, 0)
• 𝑗 = (0, 1, 0)
• 𝑘 = (0, 0, 1)

Question 13

What is the position vector of a point P(x, y, z)?

Answer 13

𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘.

Question 14

How are vectors added component-wise?

Answer 14

𝑎 + 𝑏 = (𝑎ₓ + 𝑏ₓ, 𝑎ᵧ + 𝑏ᵧ, 𝑎𝓏 + 𝑏𝓏).

Question 15

How are vectors subtracted component-wise?

Answer 15

𝑎 − 𝑏 = (𝑎ₓ − 𝑏ₓ, 𝑎ᵧ − 𝑏ᵧ, 𝑎𝓏 − 𝑏𝓏).

Question 16

What is the formula for the magnitude of a vector a?

Answer 16

|𝑎| = √(𝑎ₓ² + 𝑎ᵧ² + 𝑎𝓏²).

Question 17

What is a unit vector?

Answer 17

A vector of magnitude 1, denoted as 𝑎̂ = 𝑎 / |𝑎|.

Question 18

What is the definition of the scalar (dot) product?

Answer 18

𝑎 ⋅ 𝑏 = |𝑎||𝑏| cos(𝜃).

Question 19

What are the properties of the scalar (dot) product?

Answer 19

• Commutative: 𝑎 ⋅ 𝑏 = 𝑏 ⋅ 𝑎
• Distributive: 𝑎 ⋅ (𝑏 + 𝑐) = 𝑎 ⋅ 𝑏 + 𝑎 ⋅ 𝑐
• (𝜆𝑎) ⋅ (𝜇𝑏) = 𝜆𝜇(𝑎 ⋅ 𝑏)

Question 20

How is the dot product calculated in Cartesian coordinates?

Answer 20

For 𝑎 = (𝑎ₓ, 𝑎ᵧ, 𝑎𝓏) and 𝑏 = (𝑏ₓ, 𝑏ᵧ, 𝑏𝓏): 𝑎 ⋅ 𝑏 = 𝑎ₓ𝑏ₓ + 𝑎ᵧ𝑏ᵧ + 𝑎𝓏𝑏𝓏.

Question 21

What does a dot product of zero indicate?

Answer 21

The vectors are perpendicular.

Question 22

What is the definition of the vector (cross) product?

Answer 22

𝑎 × 𝑏 = |𝑎||𝑏| sin(𝜃) 𝑛̂.

Question 23

What is the magnitude of the vector (cross) product?

Answer 23

|𝑎 × 𝑏| = |𝑎||𝑏| sin(𝜃).

Question 24

What rule determines the direction of the vector (cross) product?

Answer 24

The right-hand rule.

Question 25

How is the vector (cross) product calculated in Cartesian coordinates?

Answer 25

𝑎 × 𝑏 = (𝑎ᵧ𝑏𝓏 − 𝑎𝓏𝑏ᵧ)𝑖 − (𝑎ₓ𝑏𝓏 − 𝑎𝓏𝑏ₓ)𝑗 + (𝑎ₓ𝑏ᵧ − 𝑎ᵧ𝑏ₓ)𝑘.

Question 26

What are the properties of the vector (cross) product?

Answer 26

• Anticommutative: 𝑎 × 𝑏 = −𝑏 × 𝑎
• Distributive: 𝑎 × (𝑏 + 𝑐) = 𝑎 × 𝑏 + 𝑎 × 𝑐
• Scalar Multiplication: 𝜆(𝑎 × 𝑏) = (𝜆𝑎) × 𝑏 = 𝑎 × (𝜆𝑏)