Dot Product

The Dot Product

A type of vector multiplication. The vectors a and b must be of the same dimensions. The dot product gives a scalar number.

Example: Find the dot product of <6,-2,0> and <5,1,4>. Apply the dot product formula (next page) : = 6*5+-2*1+0*4 = 30-2 = 28 (scalar)

Example: Find the angle between <5,-1,2,-3> and <0,6,-2,-3>. Note that these two are the same dimension, n=4. First find a dot b, then find the magnitudes |a| and |b|. a dot b = 5*0+(-1)*6+2*(-2)+(-3)*(-3) = +6-4+9 = -1 |a| = sqrt(5^2+1^2+2^2+3^2) = sqrt(39) |b| = sqrt(0^2+6^2+2^2+3^2) = sqrt(49) = 7 Apply the angle formula for dot product (next page): -1=7sqrt(39)cos(theta), cos(theta)=-1/(7sqrt(39)), theta=cos^-1(-1/7sqrt(39)) = 1.5937 rad

Example: Find the acute angles at an intersection of two functions, a parabola and line. Steps: First find where the two functions intersect. Set the two equal to each other. Simplify, set equal to 0. Find values of x by factoring. Plug in these values of x into the parabola equation to find the corresponding values of y. Find the derivative of parabola, y'. Plug in the values of x found earlier. These give the slopes of the functions at x1, x2... Find the tangent vectors for each function:       For a line: take the slope of the line from the line function. ex. for a slope of three, the tangent vector is <1,3> (1 unit over, 3 units up)       For a parabola: need tangent vectors at both points of intersection.       Find the value of y' for each x (plug in the found x values into the derivative of the parabola)       For each, this is the slope.       Turn the slope into a tangent vector like the example for a line. From here, find the angles between the tangent vectors, using the theta dot product formula (next page).  

Equations

a dot b = scalar a dot b = |a||b|*cos(theta) |a| = sqrt(a1^2+a2^2...) vector of AB = Comp_a(b)=|b|cos(theta) Comp_a(b)= ((a dot b) / |a|) Proj_a(b) = 1/|a|*(comp_a(b)a = ((a dot b)/(|a|^2))a    (Take a, shrink it by 1 over a magnitude, then stretch by a factor of comp_a(b)

Properties

Commutativity: a dot b = b dot a Distributivity: (a+b) dot c = a dot c + b dot c Norm: a dot a = |a|^2 When a dot b = 0, the angle between the two is 90 degrees.