Consider the spiral given by c(t) = (e2t cos(2t), e2t sin(2t)). Show that the angle between c and c' is constant. c'(t) = Let θ be the angle between c and c'. Using the dot product rule we have the following. c(t) · c'(t) = c(t) · c'(t) cos(θ) 2e4t = cos(θ)

angle is 45° which is constant

Step-by-step explanation:

We use formula for two vectors a and b    to calculate angle θ between them by formula

cos θ =  a . b  / magnitude of a  ×  magnitude of b

Please see the attached file

5x + 3y = - 4

step-by-step explanation:

the equation of a line in standard form is

ax + by = c ( a is a positive integer and b, c are integers )

express the line first of all in slope- intercept form

y = mx + c ( m is the slope and c the y-intercept )

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = (7, - 3) and (x₂, y₂ ) = (4, - 8)

m = = -

y = - x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, - 8 ), then

- 8 = - + c ⇒ c = -

y = - x - ← in slope-intercept form

multiply all terms by 3 to eliminate the fractions

3y = - 5x - 4 ( add 5x to both sides )

5x + 3y = - 4 ← in standard form