Calculus III Test 2 (Unit 12 and 13)

What is the formula for the angle between two vectors?

$$\Theta=\arccos\left(\frac{u\cdot v}{\left|u\right|\left|v\right|}\right)$$

What’s the formula for the vector projection of u onto v ?

$$proj_{v}u=\left(\frac{u\cdot v}{\left|v\right|}\right)\frac{v}{\left|v\right|}$$

What’s the formula for the scalar component of u in the direction of v

$$\frac{u\cdot v}{\left|v\right|}$$

How do you determine if two nonzero vectors are parallel with the cross product?

u x v = 0

What’s an important property of the cross product?

u x v = -(v x u)

What’s the area of a parallelogram equation?

$Area=$ |u x v|

What’s the equation for the area of the parallelepiped?

$$Area=\left(u\times v\right)\cdot w$$

How do you determine if a vector is coplanar?

If the triple scalar product comes out to be zero

What’s the vector equation of a line?

$$r\left(t\right)=r_{o}+vt$$

What’s the parametric equation of a line?

$$x=x_{o}+tv_1$$

$$y=y_{o}+vt_2$$

$$z=z_{o}+vt_3$$

What’s the equation for distance from a Point S to a line through P parallel to v

$$d=\frac{\left|PS\times v\right|}{\left|v\right|}$$

What is the vector equation of a plane?

$$n\cdot P_0P=0$$

What is the component equation of a plane?

$$A\left(x-x_0\right)+B\left(y-y_{o}\right)+C\left(z-z_0\right)=0$$

Distance from a point S to a plane through a point P with Normal n

$$d=\left|\frac{PS\cdot n}{\left|n\right|}\right|$$

How do you identify a cylinder?

If you’re missing one of the variables then you have a cylinder and not any other quadratic surface

What’s the general formula of an ellipsoid?

All coefficients are squared and positive on the same side.

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

What’s the general form of an elliptical paraboloid?

Two coefficients are squared and positive on one side while one unsquared term is isolated. Will open around isolated term

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$$

What’s the general form of an elliptical cone?

All coefficients are squared with one isolated term. Opens around isolated term

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}$$

What’s the general form of a hyperboloid of one sheet?

All squared terms on the same side with one negative term. Will open around the unique term

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

What’s the general form of a hyperboloid of two sheets?

All squared coefficients with two negatives. Will open around the unique term

$$\frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

What’s the general form for a hyperbolic paraboloid?

Two squared coefficients with one negative on one side while an unsquared term is isolated on another side.

$$\frac{y^2}{b^2}-\frac{x^2}{a^2}=\frac{z}{c}$$

What’s the equation for the direction of motion of the velocity vector derived from a position vector?

$$D.O.M=\frac{v}{\left\vert v\right\vert}$$

Equation for length of curve?

L=\int_{a}^{b}\!\sqrt{\left(\frac{\differentialD x}{dt}\right)^2+\left(\frac{dy}{\differentialD t}\right)^2+\left(\frac{dz}{\differentialD t}\right)^2}\,dt

Equation for unit tangent vector?

$$T=\frac{\overrightarrow{v}}{\left\vert\overrightarrow{v}\right\vert}$$

What’s the vector formula for curvature?

$$\kappa=\frac{\left\vert v\times a\right\vert}{\left\vert v\right\vert^3}$$

What’s the formula for calculating normal using unit tangent?

N=\frac{\frac{dT}{\differentialD t}}{\left\vert\frac{dT}{\differentialD t}\right\vert}