1.10 Vectors

• Vectors are commonly represented as column vectors.
• Steps to perform vector addition and subtraction using column vectors

  1️⃣ Write the vectors as column vectors

  2️⃣ Add or subtract the corresponding components

  3️⃣ Write the result as a new column vector
• The column vector representation makes vector addition and subtraction straightforward.


  True
• Match the vector concept with its description:

  Column Vector Representation ↔️ Components listed vertically

  Components ↔️ Define direction and magnitude

  Benefits ↔️ Streamlines vector operations
• What is the definition of vector addition?

  Adding two vectors
• What is the definition of vector subtraction?

  Subtracting one vector
• If the scalar is negative, scalar multiplication reverses the vector's direction.

  True
• The magnitude of a vector represents its length
• Steps to calculate a unit vector

  1️⃣ Calculate the magnitude `|u|` of the original vector `u`

  2️⃣ Divide each component of the vector `u` by `|u|` to get the components of the unit vector `û`

  3️⃣ Write the resulting unit vector `û` in column form
• What are the two key characteristics of a vector?

  Magnitude and direction
• What is the effect of scalar multiplication on a vector's direction if the scalar is negative?

  Reverses direction
• Scalar multiplication reverses the direction of a vector if the scalar is positive.

  False
• What is the resulting vector when [2, 3] is multiplied by the scalar 4?

  [8, 12]
• What is the magnitude of the 3D vector [3, 4, 5]?

  √50
• What is the unit vector of [3, 4]?

  [3/5, 4/5]
• What is the general form of a vector equation for a line?

  r = a + t(b)
• What is the 2D vector equation of a line passing through (1, 3) with direction vector [2, 1]?

  [x] = [1] + t[2]
• The column vector representation simplifies vector operations like addition and subtraction.


  True
• What is the definition of vector addition?

  Adding two vectors
• How are vectors commonly represented?

  Column vectors
• If `u = [2, 3]` and `v = [1, -1]`, then `u + v = [3, 2]` after vector addition.
• If `u = [2, 3]` and `v = [1, -1]`, then `u + v = [3, 2]` because `[2] + [1] = [3]` and `[3] + [-1] = [2]`. This is an example of vector addition
• If `u = [2, 3]` and `v = [1, -1]`, then `u - v = [1, 4]` because `[2] - [1] = [1]` and `[3] - [-1] = [4]`. This is an example of vector subtraction
• Steps to perform scalar multiplication

  1️⃣ Identify the scalar k and vector `[a]`

  2️⃣ Multiply each component of the vector by the scalar

  3️⃣ Write the resulting vector in column form
• The Pythagorean theorem is used to calculate the magnitude of a vector.


  True
• What is the magnitude of the vector `u = [3, 4]`?

  5
• A vector represented as `[a] [b]` is in column vector form.
• To perform scalar multiplication, first identify the scalar `k` and the vector
• The representation of scalar multiplication is k * [a] = [ka]
• The magnitude of a vector represents its length or size.


  True
• What is the magnitude of a unit vector?

  1
• Position vectors define the location of a point relative to the origin.

  True
• In the vector equation r = a + t(b), the variable t represents a parameter
• Vectors are used in physics to analyze forces and velocities.

  True
• What are the two key characteristics of vectors?

  Magnitude and direction
• What is the general form of a 3D vector as a column vector?

  [x] [y] [z]
• Vector subtraction involves subtracting one vector from another.
• Why is the column vector representation useful for vectors?

  Simplifies vector operations
• What is `u - v` if `u = [2, 3]` and `v = [1, -1]`?

  [1, 4]
• Vector addition and subtraction are performed component-wise.


  True