exam 1 gen notes {cal3}

Vocab

• perpendicular vs parallel
  • perpendicular: intersect at a 90-degree angle
  • parallel: same distance apart, and will never meet

Vecotrs

• Unit vector
  • magnitude of 1
• 0 vector (all 0s)
• Standard basis vector
  • All vectors have at most 1, 1 element, and rest 0s eg. <0,1,0> or <1,0,0>

Dot Product

• Magitude (length of vector)
  • $||\vec{v}||=\sqrt{a_1^2+a_2^2+\cdots +a_n^2}$
• scalar multiplication
  • $c\vec{a}$ → $<c{a}_1,c{a}_2,c{a}_n>$
• orthogonal = perpendicular
  • $\vec{a}\cdot \vec{b}=0$
  • 2 lines intersect at a right angle (dot product =0)
• Vectors are parallel
  • $\vec{a}\cdot \vec{b}=\Vert \vec{a}\Vert \Vert \vec{b}\Vert (\theta =0^{\circ })\text{OR}\vec{a}\cdot \vec{b}=-\Vert \vec{a}\Vert \Vert \vec{b}\Vert (\theta =180^{\circ })$
• Dot/Scalar Product
  • $\vec{a}\ast \vec{b}=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$
  • geoMetric interp: $\Vert \vec{a}\Vert \Vert \vec{b}\Vert \cos \theta$
• Projections
  • Vector Projection: ${\text{proj}}_{\vec{a}}\vec{b}=\frac{\vec{a}\cdot \vec{b}}{\Vert \vec{a}{\Vert }^2}\vec{a}$
    • or dotProduct/mag(a) ^2 $\ast \vec{a}$
    • Where project b unto a
  • Scalar Projection: $pro{j}_a\vec{b}=\frac{\vec{a}\cdot \vec{b}}{||\vec{b}||}$
    • project a unto b → dot / mag(a)
• ONCS:
  • Steps:
    • Normalize vector (div e/a component by magnitude)
    • Find perpendicular
      • For 2d, flip, negate if needed
      • 3d, take cross product of any vector not parallel to $\vec{v}$
• Linear combo for certain $hat$ vector
  • Add each vector’s $hat$ +

Formulas

• Distance btwn points in 3d cords system ?
  • $\sqrt{(x_2-x_1{)}^2-(y_2-y_1{)}^2-(z_2-z_1{)}^2}$ adding as many planes as needed
• Find angle btwn 2 vectors
  • $\cos (\theta )=\frac{u\cdot v}{|u||v|}$
• Find values of x for angle between vector = 45deg
  • $\vec{v}\vec{\cdot }w=||v||\cdot ||w||\cdot \cos (45)$
  • for v&w being some vector
• Find point to parametric line r(t) = t$\vec{a}$ and $\vec{a}$
  • $d=\frac{||\vec{v}\vec{\times }a||}{||\vec{a}||}$
  • distnace = mag(cross) / mag of a
• formula for the length of the a parametric curve cal
  • ${\int }_a^b||r^{\prime }(t)||dx$
  • (integral of mag of derative)
• Area of triangle
  • $\frac{1}{2}|\vec{v}\times \vec{w}|$
    • Normally magnitude of cross product of both vectors
• magnitude of the cross product
  • area of the parallelogram formed by those two vectors

Cross Pproduct

• Cross Product ch 11.4 {cal3}
• What does $\vec{a}\vec{\times }b=\vec{0}$ tell us about both vectors?
  • then both vectors are parallel vectors
• What does $\vec{a}\vec{\times }b\ne \vec{0}$ tell us abotu both vecotrs?
  • cross product is orthogonal to both vectors

Determinent