1.10 Vectors

• What does the magnitude of a vector represent?

  The length or size
• What happens to a vector's direction when multiplied by a scalar?

  Changes only if scalar is negative
• How are vectors denoted in vector notation?

  Bold lowercase letters
• What is the component-wise method for vector addition?

  Add corresponding components
• What happens to a vector's direction when multiplied by a negative scalar?

  It reverses
• The magnitude of a vector is often denoted as |v|
• The magnitude of a vector is its length or size


  True
• Multiplying a vector by a scalar changes its direction

  False
• What is the formula for calculating the magnitude of a vector using its components?

  $|v| =$$√(x^{2} +$$y^{2} +$$z^{2})$
• What does scalar multiplication change in a vector?

  Magnitude
• What does a position vector specify?

  Location of a point
• What is the dot product of two vectors also called?

  Scalar product
• Position vectors define the location of a point, while displacement vectors describe the change in position between two points.


  True
• The formula for the dot product is a ⋅ b = |a||b|cos(θ), where θ represents the angle
• Steps to calculate the angle between two vectors using the dot product

  1️⃣ Calculate the dot product of the vectors

  2️⃣ Calculate the magnitudes of the vectors

  3️⃣ Use the formula θ = cos^{-1}(a ⋅ b / |a||b|)
• What is the formula for the cross product of two vectors?

  \mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin(\theta)\mathbf{n}</latex>
• In what areas is the cross product useful?

  Normal vector to a plane
• Scalar multiplication of a vector changes its magnitude but not its direction.


  True
• The magnitude of the vector (3, 4, 2) is √29.


  True
• In geometric vector addition, vectors are placed "tip-to-tail".
• In the geometric method of vector addition, vectors are placed end-to-end
• What does scalar multiplication change about a vector?

  Magnitude
• Multiplying a vector by a positive scalar stretches it, while multiplying by a negative scalar stretches and flips it over.

  True
• Match the vector property with its description:

  Position Vector ↔️ Specifies the location of a point in space

  Displacement Vector ↔️ Represents the change in position
• In the dot product formula, θ represents the angle between the two vectors
• What formula is used to calculate the angle between two vectors using the dot product?

  \theta = \cos^{ - 1}\left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)</latex>
• The dot product of two vectors results in a scalar quantity.

  True
• The distributive property of the dot product is expressed as a ⋅ (b + c) = a ⋅ b + a ⋅ c
• The dot product of two vectors is equal to |a||b|cos(θ).


  True
• The cross product is anticommutative, meaning a × b = -b × a.


  True
• Steps to calculate the cross product using vector components:

  1️⃣ Identify the components of the two vectors a and b

  2️⃣ Apply the cross product formula: (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

  3️⃣ Simplify the resulting vector
• Vectors expressed in parametric form use a parameter often denoted as t
• Match the geometric quantity with its description:

  Displacement ↔️ Change in position

  Velocity ↔️ Rate of change of position

  Force ↔️ Push or pull on an object
• How do you add two vectors using their components?

  Add corresponding components
• What is a vector defined as?

  A mathematical quantity with magnitude and direction
• Vectors can be added by placing them end-to-end
• The length or size of a vector is called its magnitude
• Vectors expressed in terms of their x, y, and z components are using vector components.
• In the geometric method for vector addition, the tail of the second vector is placed at the tip of the first vector.

  True
• Multiplying a vector by a positive scalar stretches it, while multiplying by a negative scalar stretches and flips it over.


  True