Vectors

Last updated Jun 15, 2023 Edit Source

# ← Specialist Home

• Cambridge Vectors
• Sadler Vectors
• Bajay’s ClassPad Guide
• Vectors with Noah

• $\vec{AB}$ - A vector
• $\vec{|AB|}$ - The magnitude of a vector
• $\overline{AB}$ - A line

• $\vec{AB} + \vec{BC} = \vec{AC}$

  • Adjacent, identical letters cancel each other out
    • e.g: $\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} = \vec{AE}$

• # Addition of Vectors Geometrically

  • Two vectors $\vec{u}$ and $\vec{v}$ can be added together by drawing a line segment representing u from A to B and then a line segment representing v from B to C
  • Using cosine rule we can find the magnitude of $\vec{u}$ + $\vec{v}$
    • $c^{2}=a^{2}+b^{2}-2ab \times cos(C)$
  • Using sine rule we can find the angles of the triangle these vectors form
    • $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$
    • This can allow us to find the direction of $\vec{u}$ + $\vec{v}$

• $\vec{AB} -\vec{CB} = \vec{AC}$
  • Rewrite negative vectors by reversing direction
    • e.g: $\vec{-CB} = \vec{BC}$
    • Thus $\vec{AB} - \vec{CB} = \vec{AB} + \vec{BC} = \vec{AC}$
      • It can be solved using vector addition

• ABCD is a parallelogram:
• Lines of the same length are still different line segments as they have different coordinates
  • E.g: $\overline{AB} \neq \overline{DC}$
• However vectors of the same magnitude are the same
  • E.g: $\vec{AB} = \vec{DC}$
  • Vectors don’t have a fixed location (But the things they describe might)
  • A vector is a number representation of the distance between two points

• Vectors can be represented abstractly as lowercase letters from either:
  • having a tilde underneath - e.g: $\underset{\sim}{a}$
    • For the sake of clarity, most vectors have a tilde underneath them, however from unit vectors onward, they will just be represented as $a$
  • being underlined - e.g: $\underline{a}$
  • or being in bold - e.g: a or $a$

• # Mathematical Rules for Abstract Vectors

  • # Equality

    • Vectors are the same if and only if their magnitudes ($|a|=|b|$) are equal and their directions are equal
      • Vectors with same magnitude and direction are “like” vectors
      • Vectors with same magnitude and different directions are “unlike” vectors
      • Vectors with different magnitude and same direction are “like” vectors
      • 2 vectors that are parallel, but are in opposite direction are “unlike” vectors

  • # The Negative of a Vector

    • For a vector $\underset{\sim}{a}$, the vector $\underset{\sim}{-a}$ has the same magnitude but opposite direction
    • $\vec{-AB} = \vec{BA}$

  • # Scalar Multiplication

    • For a vector ($\underset{\sim}{a}$) and positive scalar (${k}$), the vector $\underset{\sim}{a}$ is the vector with the same direction as $\underset{\sim}{a}$ and same magnitude as ${k|a|}$
      • Given for vector $\underset{\sim}{a}$, → $\underset{\sim}{2a} = \underset{\sim}{a} + \underset{\sim}{a}$
      • For when ${k}$ is negative, ${k}\underset{\sim}{a}$ = ${k|-a|}$

  • # Subtracting Vectors

    • $\underset{\sim}{a} - \underset{\sim}{b}$ = $\underset{\sim}{a} + \underset{\sim}{(-b)}$

  • # The Zero Vector

    • The zero vector 0 has magnitude 0 and an undefined direction

• 2D Vectors can be represented as a sum of a horizontal ${(i)}$ and vertical ${(j)}$
  • Each vector $\underset{\sim}{u}$ in the plane can be written in component form as $u = xi + yj$, where:
    • $i$ is the unit vector in the positive direction of the $x$-axis
    • $j$ is the unit vector in the positive direction of the $y$-axis
  • Vector $\underset{\sim}{a}$ can be written in component form → $\underset{\sim}{a}$ = $|a|\\ \cos(\theta)\\ i + |a|\\ \sin(\theta)\\ j$
  • Vectors in component form are equal if and only if their components are equal:
    • $ai + bj = ci + dj$ if and only if $a = c$ and $b = d$
• Vectors in component form can be added by adding their components and simplifying
  • ${ai + bj + ci + dj = (a+c)i + (b + d)j}$
• From component form, the magnitude of a vector can be found through pythagoras theorem
  • For $\underset{\sim}{x} = ai + bj$
    • $|x| = \sqrt{a^2 + b^2}$

• A unit vector is a vector of length one unit
  • Unit vectors are represented with a ‘hat’ → ${\hat{u}}$
  • ${\hat{u}} = \frac{u}{|u|}$

• Vector used to represent the position of a point relative to the origin (in the Cartesian plane when working in a 2D space)
  • $\vec{AB} = −O\vec{A} + O\vec{B}$
    • $O$ represents the origin

• The scalar product (also known as dot product) is an operation that takes two vectors and gives a real number
• $a \cdot b = a_1b_2 + a_2b_2$
  • E.g: $a = 2i + 3j, b = i - 4j$
  • $a \cdot b = (2 \\ × \\ 1) + (3 \\ × -4) = -10$
• $a \cdot b = |a||b|\cos(\theta)$
• $\cos(\theta) = \frac{a\\ \cdot \\ b}{|a||b|} = \frac{a_1b_1\\ + \\ a_2b_2}{|a||b|}$
  • This equation allows us to find the angle between two vectors
  • $\theta$ is the angle between vectors arranged tail-to-tail or nose-to-nose
• If $a$ or $b = 0$, then $a \cdot b = 0$
• Rules for scalar products:
  • $a \\ \cdot \\ b = b \\ \cdot \\ a$
  • $a \\ \cdot \\ (b\\ + \\ c) = a\\ \cdot \\ b \\ +\\ a \\ \cdot \\ c$
  • $\lambda(a\\ \cdot \\ b) = a\\ \cdot \\ (\lambda b) = (\lambda a) \\ \cdot \\ b$
  • $a \\ \cdot \\ a = |a|^2$
  • $a$ and $b$ are perpendicular if and only if $a\\ \cdot \\ b = 0$
  • $a\\ \cdot \\ b = |a| |b|$ if $a$ and $b$ are parallel and in the same direction
  • $a\\ \cdot \\ b = -|a| |b|$ if $a$ and $b$ are parallel and in the opposite direction

• # Scalar Projections

  • The scalar projection of $a$ onto $b$ is the scalar component of the shadow of $a$ on $b$
  • |$a$| $\cos(\theta)$
    • If $\theta$ is obtuse then $\cos(\theta)$ is negative
      • The scalar projection will be negative
  • The original equation can be rearranged to → $\frac{a\\ \cdot\\ b}{|b|}$

• # Vector Projections

  • The vector projection is the vector representing the shadow of $a$ on $b$
    • Thus it will have a direction/be represented in column form
  • $(a\\ \cdot \\ \hat{b})\hat{b}$ → projection of $a$ onto $b$
    • Also can be rewritten as → $\frac{a\\ \cdot\\ b}{|b|^2} \cdot b$
    • Solve the unit vector first then solve the equation
  • |$a$| $\cos(\theta)\\ \hat{b}$ → Alternative formula
    • $\hat{b}$ is the unit vector in the direction of $\hat{b}$

• Vector Algebra Rules:
  • $a = b$ if and only if $a$ and $b$ have the same magnitude and direction
  • $-a$ has the same magnitude as $a$ but the opposite direction
  • For a scalar $\lambda >0$, $\lambda {a}$ has magnitude $\lambda |a|$
  • If $a = \lambda{b},$ and $\lambda > 0$, $a$ and $b$ are like parallel vectors
    • Otherwise, if $\lambda<0$, $a$ and $b$ are unlike vectors
  • Vectors can be added with a triangle of vectors
  • $a − b = a + (−b)$
  • f $\lambda$ is a scalar, then $\lambda(a + b) = \lambda{a} + \lambda{b}$
  • $a$ and $b$ are perpendicular if and only if $a\\ × \\ b = 0$

• Using the concept of position vectors:
  • The position of vector $a$ relative to vector $b$ → $a - b$
  • The velocity of vector $a$ relative to vector $b$ → $-a + b$
  • e.g: The velocity of ship A, as seen by an observer on ship B, is 7i - 10j. The velocity of ship A, as seen by an observer on ship C, is 13i - 2j. Find the velocity of ship B as seen by an observer on C.
    • A$r$B = 7i - 10j | A$r$B = $r$A - $r$B
    • A$r$C = 13i - 2j | A$r$C = $r$A - $r$C
    • B$r$C = ? | B$r$C = $r$B - $r$C
    • $r$B - $r$C = $r$A - $r$C - ($r$A - $r$B)
      • = 13i - 2j - (7i - 10j)
      • 6i + 8j km/h