1 Vectors and Geometry Free Online FlashCards

How do you find the vector from $P$ to $Q$?

For points P and Q, subtract coordinates componentwise: $\overrightarrow{PQ}=Q-P=\langle q_1-p_1,q_2-p_2\rangle$ in two dimensions, with the analogous three-dimensional formula.

How does scalar multiplication affect a vector?

Scalar multiplication is componentwise: $c\langle v_1,v_2,v_3\rangle=\langle cv_1,cv_2,cv_3\rangle$. For $c<0$, the vector reverses direction; its magnitude is multiplied by $|c|$.

What is a linear combination?

A linear combination has the form $c_1\mathbf v_1+\cdots+c_k\mathbf v_k$, where the $c_i$ are scalars.

What is the Euclidean norm of a vector?

The Euclidean norm is $\|\mathbf v\|=\sqrt{v_1^2+v_2^2+\cdots+v_n^2}$. It measures the vector’s magnitude or length.

What is the component formula for a dot product?

The dot product is $\mathbf u\cdot\mathbf v=u_1v_1+\cdots+u_nv_n$. Its result is a scalar, not a vector.

What is the vector projection of $\mathbf v$ onto $\mathbf u$?

The vector projection is $\operatorname{proj}_{\mathbf u}\mathbf v=\frac{\mathbf v\cdot\mathbf u}{\|\mathbf u\|^2}\mathbf u$, for nonzero $\mathbf u$. It gives the part of $\mathbf v$ parallel to $\mathbf u$.

What is the vector equation of a line?

A line through position vector $\mathbf r_0$ with nonzero direction vector $\mathbf d$ is $\mathbf r(t)=\mathbf r_0+t\mathbf d$, where $t\in\mathbb R$.

How is a plane written using a point and normal vector?

A plane through $P_0=(x_0,y_0,z_0)$ with normal $\mathbf n=\langle A,B,C\rangle$ satisfies $\mathbf n\cdot(P-P_0)=0$, or $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$.

What does a two-direction vector equation describe?

The equation $\mathbf r=\mathbf r_0+s\mathbf u+t\mathbf v$ describes a plane when $\mathbf u$ and $\mathbf v$ are not parallel. Two independent directions allow movement across the plane.

What is a unit vector?

A vector with magnitude $1$ is a unit vector. In $\mathbb{R}^3$, the standard unit vectors are $\mathbf i=\langle1,0,0\rangle$, $\mathbf j=\langle0,1,0\rangle$, and $\mathbf k=\langle0,0,1\rangle$.

What does the span of vectors mean?

The span of vectors is the set of all their linear combinations. For example, $\operatorname{span}\{\mathbf u,\mathbf v\}=\{s\mathbf u+t\mathbf v:s,t\in\mathbb R\}$.

When are vectors linearly independent?

Vectors are linearly independent when $c_1\mathbf v_1+\cdots+c_k\mathbf v_k=\mathbf0$ has only the trivial solution $c_1=\cdots=c_k=0$.