Calc 3 Vector Basics Masterclass Flashcards

Vocabulary practice flashcards covering fundamental 2D and 3D vector concepts, operations, dot products, projections, and cross products for Calculus 3.

Scalar

A real number that has only magnitude (size) and no direction, such as temperature ($72^\circ\text{F}$) or mass ($5\,\text{kg}$).

Vector

A mathematical object characterized solely by having both magnitude (length) and direction, represented geometrically as an arrow from an initial point $P$ to a terminal point $Q$.

Standard Position

The placement of a vector on a coordinate grid such that its initial point (tail) is located at the origin $(0, 0)$ and its terminal point (tip) is at $(v_1, v_2)$.

Component Form

The representation of a vector using angle brackets, written as $\langle v_1, v_2 \rangle$, where $v_1$ is horizontal displacement and $v_2$ is vertical displacement.

Magnitude

The length of a vector $v = \langle v_1, v_2 \rangle$, denoted as $||v||$ or $|v|$, calculated using the Pythagorean theorem formula $||v|| = \sqrt{v_1^2 + v_2^2}$.

Tip-to-Tail Rule

The geometric procedure for vector addition where the tail of vector $v$ is placed at the tip of vector $u$, resulting in $u + v$ running from the tail of $u$ to the tip of $v$.

Scalar Multiplication

The operation of multiplying a vector by a scalar $c$, defined algebraically as $c\langle v_1, v_2 \rangle = \langle cv_1, cv_2 \rangle$, which scales vector length and reverses direction if $c < 0$.

Unit Vector

Any vector whose magnitude (length) is equal to exactly $1$.

Standard Basis Vectors (2D)

The unit vectors along the coordinate axes in 2D space, defined as $i = \langle 1, 0 \rangle$ along the x-axis and $j = \langle 0, 1 \rangle$ along the y-axis.

Vector Normalization

The process of constructing a unit vector $u$ in the exact same direction as a non-zero vector $v$ by dividing $v$ by its magnitude: $u = \frac{v}{||v||}$.

Standard Basis Vectors (3D)

The standard unit vectors in 3D space, defined as $i = \langle 1, 0, 0 \rangle$, $j = \langle 0, 1, 0 \rangle$, and $k = \langle 0, 0, 1 \rangle$.

Right-Hand Rule for Axes

An orientation rule for 3D coordinate systems where pointing the right index finger along the positive x-axis and curling fingers toward the positive y-axis results in the thumb pointing toward the positive z-axis.

3D Distance Formula

The formula used to calculate distance between points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Coordinate Planes in R³

Planes formed by fixing one coordinate to zero: $z = 0$ defines the xy-plane, $y = 0$ defines the xz-plane, and $x = 0$ defines the yz-plane.

Standard Equation of a Sphere

The algebraic equation $(x - h)^2 + (y - k)^2 + (z - l)^2 = R^2$, representing all points at constant distance $R$ from center $(h, k, l)$.

Dot Product

An operation combining two vectors to produce a scalar, defined algebraically as $u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$ and geometrically as $u \cdot v = ||u|| ||v|| \cos(\theta)$.

Orthogonal Vectors

Two non-zero vectors that are perpendicular to each other, occurring if and only if their dot product is equal to zero ($u \cdot v = 0$).

Scalar Component (comp_v u)

The length or magnitude of the shadow created by vector projection of $u$ onto $v$, given by $\text{comp}_v u = \frac{u \cdot v}{||v||}$.

Vector Projection (proj_v u)

The actual shadow vector of $u$ directed along $v$, calculated using the formula $\text{proj}_v u = \left(\frac{u \cdot v}{||v||}\right) \frac{v}{||v||}$.

Cross Product

An operation defined exclusively in 3D space that combines two vectors $u$ and $v$ to produce a new vector perpendicular to both original vectors.

Anti-commutative Property

A mathematical property of the cross product where changing the order of multiplication negates the resulting vector: $v \times u = -(u \times v)$.

Cross Product Magnitude Formula

The magnitude of a cross product given by $||u \times v|| = ||u|| ||v|| \sin(\theta)$, which equals the area of the parallelogram formed by vectors $u$ and $v$.

Parallel Test (Cross Product)

A test determining that two vectors $u$ and $v$ are parallel if and only if their cross product yields the zero vector ($u \times v = \mathbf{0}$).

Scalar Triple Product

A $3 \times 3$ determinant expression combining vectors $u$, $v$, and $w$ used to calculate the volume of a 3D box (parallelepiped).