01 Vectors and Geometry in Space Free Online FlashCards

What distinguishes a vector from a point?

A vector has both magnitude and direction, whereas a point identifies only a location.

How do you calculate the vector from P to Q?

What is the position vector of $P=(x,y,z)$?

The position vector of $P=(x,y,z)$ is $\overrightarrow{OP}=\langle x,y,z\rangle$.

How are vectors added or subtracted?

Add or subtract corresponding components: $\langle a,b,c\rangle\pm\langle d,e,f\rangle=\langle a\pm d,b\pm e,c\pm f\rangle$.

What is the magnitude of a vector in $\mathbb{R}^3$?

For $\mathbf{v}=\langle v_1,v_2,v_3\rangle$, $\|\mathbf{v}\|=\sqrt{v_1^2+v_2^2+v_3^2}$.

What does the dot product produce?

The dot product is $\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2+u_3v_3$, and it produces a scalar.

What dot-product test identifies perpendicular vectors?

If $\mathbf{u}\cdot\mathbf{v}=0$, the vectors are perpendicular, assuming neither vector is zero.

How is scalar projection of $\mathbf{u}$ onto $\mathbf{v}$ computed?

The scalar projection of $\mathbf{u}$ onto $\mathbf{v}$ is $\operatorname{comp}_{\mathbf{v}}\mathbf{u}=\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|}$.

What kind of object is a cross product?

The cross product produces a vector perpendicular to both input vectors in $\mathbb{R}^3$.

How do you find the triangle area from two edge vectors?

The area is $A=\frac{1}{2}\|\mathbf{u}\times\mathbf{v}\|$.

What determines a line in space?

A line through $P_0$ with direction vector $\mathbf{v}$ has vector equation $\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}$.

What are skew lines?

Skew lines are neither parallel nor intersecting.