Vectors

• What is a vector?

  A vector is a quantity that has both magnitude and direction.
• How can vectors be described?

  Vectors can be described as directional growth represented by lines that can vary in size.
• How is a vector typically represented?

  A vector is usually represented on a coordinate grid, but it doesn't have to be.
• How can we label a vector?

  A vector can be labeled as A to B or as vector c, usually underlined or bolded when typed.
• What are the components of a vector?

  Vectors can be expressed in component form, usually as x, y, and z coordinates.
• How do you interpret the vector (2, 7)?

  The vector (2, 7) means to move 2 units in the x direction and 7 units in the y direction.
• How do you find the magnitude of a two-dimensional vector?

  The magnitude of a two-dimensional vector is found using the Pythagorean theorem: $\sqrt{a^2 + b^2}$.
• How is the magnitude of a three-dimensional vector calculated?

  The magnitude of a three-dimensional vector is calculated as $\sqrt{a^2 + b^2 + c^2}$.
• What notation is used to denote the magnitude of a vector?

  The magnitude of a vector is denoted by $|U|$.
• When are two vectors considered equal?

  Two vectors are equal if they have the same magnitude and direction, regardless of their position.
• How can you demonstrate that two vectors are equal using components?

  If vector A has components (a, b, c) and vector B has components (p, q, r), then A equals B if $a =$$p$, $b =$$q$, and $c =$$r$.
• How do you add vectors?

  You add vectors by adding their corresponding x, y, and z components together.
• If vector U is (8, -2, 0) and vector V is (4, 6, -3), what is U + V?

  U + V is (12, 4, -3).
• How can you visualize the addition of two vectors?

  You can visualize vector addition by placing the tail of one vector at the head of the other and drawing the resultant vector from the tail of the first to the head of the second.
• How do you subtract vectors?

  Subtracting vectors is equivalent to adding the negative of the vector you want to subtract.
• If vector U is (8, -2, 0) and vector V is (4, 6, -3), what is U - V?

  U - V is (4, -8, 3).
• What does multiplying a vector by a scalar mean?

  Multiplying a vector by a scalar means to multiply each component of the vector by that scalar.
• If vector U is (8, -2, 0), what is 3U?

  3U is (24, -6, 0).
• What happens when you multiply a vector by a negative scalar?

  Multiplying a vector by a negative scalar reverses its direction.
• What is a position vector?

  A position vector is a vector from the origin to a specific point.
• How can you express the position vector of point A?

  The position vector of point A can be expressed as vector OA, which goes from the origin to point A.
• How can you find the vector from point A to point B using position vectors?

  The vector from A to B can be found using the formula: vector AB = vector OA + vector OB.
• What is the formula for finding the vector from point A to point B using their position vectors?

  The formula is vector AB = vector OB - vector OA.
• If point P has coordinates (5, -6, 9) and point Q has coordinates (3, 2, 5), what is the position vector from P to Q?

  The position vector from P to Q is (-2, 8, -4).
• How do you find the distance between two points using their position vectors?

  The distance between two points can be found by calculating the magnitude of the vector connecting them.
• If the position vectors of points P and Q are (5, -6, 9) and (3, 2, 5) respectively, what is the distance between P and Q?

  The distance between P and Q is $2\sqrt{21} \text{ units}$.
• What does it mean for two lines to be collinear in vector terms?

  Two lines are collinear if they are parallel and share a common point.
• How can you determine if two vectors are parallel?

  Two vectors are parallel if they have the same direction, regardless of their magnitudes.
• How can you check if two vectors are parallel using their components?

  You can check if two vectors are parallel by examining the ratios of their corresponding components.
• If vector A is (12, -3, 9) and vector B is (20, -5, 15), are they parallel?

  Yes, vector B is a multiple of vector A, indicating they are parallel.
• How can you determine if two vectors are parallel using the ratio method?

  You can determine if two vectors are parallel by simplifying the ratios of their corresponding components and checking if they are equal.
• What does it mean when two vectors are parallel?

  It means they are going in the same direction.
• If vector V is double the size of vector U, how can we express this relationship mathematically?

  V = 2U
• How can we determine if two vectors are parallel using their components?

  By examining the ratios of their corresponding components.
• What is the first step to check if vectors A and B are parallel?

  Examine the ratio between each of their components.
• What are the components of vector A in the example given?

  12, -3, and 9.
• What are the components of vector B in the example given?

  20, -5, and 15.
• How can we simplify the vectors to check for parallelism?

  By dividing each component by their common factors.
• What is the process to show that three points A, B, and C are collinear?

  • Show that vector A to B is a multiple of vector B to C.
  • Calculate the position vectors and check their ratios.
• If A to B is equal to half of B to C, what can we conclude about the segments?

  They are parallel with B as a common point.