3.11 J: Vectors

• The components of a vector describe it in terms of its x and y coordinates.

  True
• How do you subtract the horizontal components of two vectors?

  Subtract them
• What is the result of subtracting vector b from vector a if their components are subtracted as a_x - b_x and a_y - b_y?

  (a_x - b_x, a_y - b_y)
• How do you add two vectors using components?

  Add x and y components
• If a = (3, 2) and b = (1, -4), what is a - b?

  (2, 6)
• The magnitude of a vector is its length.


  True
• If a = (4, 3), what is the direction of a in degrees?

  36.9°
• What are the two components of a vector called?

  Horizontal and vertical
• Steps to add two vectors using components

  1️⃣ Add the horizontal components: a_x + b_x

  2️⃣ Add the vertical components: a_y + b_y

  3️⃣ The result is the new vector (a_x + b_x, a_y + b_y)
• What is the result of multiplying a vector by a scalar k?

  (k*a_x, k*a_y)
• The direction of a vector is the angle it makes with the positive x-axis.


  True
• The dot product of two vectors can be used to calculate the angle between them.
• The dot product of two vectors results in a scalar value.


  True
• What type of result does the cross product yield?

  A vector
• What is the magnitude of the cross product of two vectors `a` and `b`?

  $|a||b|sin(θ)$
• If `a = (3, 2, 1)` and `b = (1, -1, 2)`, what is the cross product `a × b`?

  (5, -7, -5)
• The advantages of using matrix representation for vectors include compact notation and the ability to perform operations easily.
• The horizontal and vertical parts of a vector are called its components.
• What is the formula for calculating the magnitude of a vector `a = (a_x, a_y)`?

  $|a| =$ \sqrt{a_{x}^{2} + a_{y}^{2}}
• Steps to find the angle between two vectors using the dot product

  1️⃣ Calculate the dot product

  2️⃣ Find the magnitudes

  3️⃣ Solve for cos θ

  4️⃣ Take the inverse cosine
• The dot product results in a scalar value.


  True
• The magnitude of the cross product involves sin θ.


  True
• One advantage of using matrix form for vectors is that operations become easier for computation
• What is the process for adding two vectors `a = (a_x, a_y)` and `b = (b_x, b_y)`?

  $(a_{x} +$$b_{x}, a_{y} +$$b_{y})$
• What types of quantities can vectors represent, as mentioned in the study material?

  Forces or displacements
• What is the first step in adding or subtracting two vectors `a = (a_x, a_y)` and `b = (b_x, b_y)`?

  Add/subtract horizontal components
• Knowing the magnitude and direction of a vector allows us to fully describe it.


  True
• The components of a vector are its horizontal and vertical parts.
• When adding vectors, the vertical components are added to find the new vector's vertical component.
• What is the result of adding vectors a and b if their components are added as a_x + b_x and a_y + b_y?

  (a_x + b_x, a_y + b_y)
• The components of a vector describe it in terms of its x and y coordinates.
• If a = (3, 2) and b = (1, -4), what is a + b?

  (4, -2)
• If a = (3, 2) and k = 4, what is k*a?

  (12, 8)
• If a = (4, 3), what is the magnitude of a?

  5
• Vectors have both magnitude and direction.
• The horizontal and vertical components of a vector are denoted as a_x and a_y.
• Match the vector operation with its calculation:

  Addition ↔️ a_x + b_x, a_y + b_y

  Subtraction ↔️ a_x - b_x, a_y - b_y
• How is the magnitude of a vector a = (a_x, a_y) calculated?

  $|a| =$ \sqrt{a_{x}^{2} + a_{y}^{2}}
• Steps to find the direction of a vector

  1️⃣ Calculate tan θ = a_y / a_x

  2️⃣ Use tan^-1 to find θ

  3️⃣ Adjust the angle if necessary
• Steps to calculate the angle between two vectors using the dot product

  1️⃣ Calculate the dot product: a · b = a_x * b_x + a_y * b_y

  2️⃣ Find the magnitudes: |a| and |b|

  3️⃣ Solve for cos θ: (a · b) / (|a| |b|)

  4️⃣ Take the inverse cosine: θ = cos^-1((a · b) / (|a| |b|))