MA 162 Exam 1

Angle between vectors (sin)

$$\sin=\frac{\vert axb\vert}{\left\Vert a\right\Vert\left\Vert b\right\Vert}$$

angle between vectors (cos)

$$\cos=\frac{a\cdot b}{\left\Vert a\right\Vert\left\Vert b\right\Vert}$$

Disk/washer

$$V=\pi\int\left(R\right)^2dx$$

Shell

$$V=2\pi\int xR\left(x\right)dx$$

Volume of solid with cross section

$$\displaylines{V=\int\left(CrossSectionArea\right)dy}$$

Surface Area

$$SA=2\pi\int y\sqrt{1+\left(y^{\prime}\right)^2}dx$$

dot product

$$u\cdot v=\left\vert u\right\vert\left\vert v\right\vert\cos$$

$$u\cdot v=u1v1+u2v2$$

$$\left\vert v\right\vert^2=v\cdot v$$

if orthogonal: $u\cdot v=0$

Orthogonal projections

$\text{proj}<em>{v}u = |u| \cos(\theta) \left(\frac{v}{|v|}\right) \dots$
$= \text{scal}</em>{v}u \left(\frac{v}{|v|}\right) = \left(\frac{u \cdot v}{v \cdot v}\right)v$

Scalar component of u in direction of v

$$scal_{v}u=\left\vert u\right\vert\cos=\frac{u\cdot v}{\left\vert v\right\vert}$$

Magnitude of cross product

$$\left\vert u\times v\right\vert=\left\vert u\right\vert\left\vert v\right\vert\sin$$