Calc 3 FlashCards October

Distance between two points in 3D

$$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Magnitude of a vector

$$\sqrt{a^2 + b^2 + c^2}$$

Unit vector

$$\frac{\mathbf{v}}{|\mathbf{v}|}$$

Dot product

$$a_1 b_1 + a_2 b_2 + a_3 b_3$$

Uses of dot product

Determining angles, testing orthogonality, and calculating vector projections.

Angle between vectors

$$\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$$

Cross product

$\mathbf{u} \times \mathbf{v}$, which yields a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$

Area of parallelogram

$$|\mathbf{u} \times \mathbf{v}|$$

Area of triangle

$$\frac{1}{2} |\mathbf{u} \times \mathbf{v}|$$

Volume of parallelepiped

$$|\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|$$

Point-to-plane distance

$$\frac{|A x_0 + B y_0 + C z_0 - D|}{\sqrt{A^2 + B^2 + C^2}}$$

Plane from point and normal

$$A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$$

Plane through three points

Take the cross product of two vectors in the plane to find a normal vector $\mathbf{n}$, then apply the point-normal form.

Parametric line

$$\mathbf{r}(t) = P + t \mathbf{v}$$

Derivative of vector function

Differentiate each component function individually.

Velocity

$\mathbf{v}(t) = \mathbf{r}'(t)$, the first derivative of position with respect to time.

Acceleration

$\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$, the second derivative of position with respect to time.

Speed

$|\mathbf{v}(t)|$, the magnitude of the velocity vector.

Arc length

$$\int_a^b |\mathbf{v}(t)|\,dt$$

Curvature

$$\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}$$

Tangential acceleration

$$a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}$$

Normal acceleration

$$a_N = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|}$$

Cylindrical coordinates

$x = r \cos(\theta)$, $y = r \sin(\theta)$, $z = z$

Cylindrical identity

$$x^2 + y^2 = r^2$$

Cylindrical volume element

$$dV = r\,dz\,dr\,d\theta$$

Spherical coordinates

$x = \rho \sin(\phi) \cos(\theta)$, $y = \rho \sin(\phi) \sin(\theta)$, $z = \rho \cos(\phi)$

Spherical identity

$$x^2 + y^2 + z^2 = \rho^2$$

Cylindrical to spherical

$r = \rho \sin(\phi)$, $z = \rho \cos(\phi)$

Point-to-line distance

$\frac{|\mathbf{v} \times \mathbf{w}|}{|\mathbf{v}|}$, where $\mathbf{v}$ is the direction of the line and $\mathbf{w}$ is a vector connecting the line to the point.

Intersection of two planes

Solve both plane equations simultaneously as a system to obtain the parametric equation of the line.