Flashcards: Vectors and Vector Subspaces — 43 cards

What is the real plane $\mathbb{R}^2$ defined as?

The set of ordered pairs of real numbers.

What is the real space $\mathbb{R}^3$ defined as?

The set of ordered triples of real numbers.

What magnitude does the geometric vector $\overrightarrow{AB}$ have?

The magnitude of segment AB.

What direction does the geometric vector $\overrightarrow{AB}$ have?

The direction of line AB.

What orientation does the geometric vector $\overrightarrow{AB}$ have?

The orientation from A to B.

When are two vectors considered equal?

When they have the same magnitude, direction, and orientation.

How is the sum of two nonzero vectors with different directions represented?

By the diagonal of the parallelogram formed by those vectors as sides.

What effect does a positive scalar λ have on a vector's orientation and magnitude?

It preserves orientation and multiplies magnitude by λ.

What effect does a negative scalar λ have on a vector's orientation?

It reverses the vector's orientation.

What is the result of multiplying any vector by zero?

The zero vector.

What defines a vector as a linear combination of two vectors?

It can be expressed as $\vec w=\lambda\vec u+\mu\vec v$ for real numbers λ and μ.

What is the standard basis of $\mathbb{R}^2$ in an orthonormal frame?

$(\vec i, \vec j)$

What is the standard basis of $\mathbb{R}^3$ in an orthonormal frame?

$(\vec i, \vec j, \vec k)$

What is the formula for the magnitude of a vector $(x,y)$ in $\mathbb{R}^2$?

$\|\vec u\| = \sqrt{x^2 + y^2}$

What is the coordinate vector $\overrightarrow{AB}$ in $\mathbb{R}^2$?

$(x_B - x_A, y_B - y_A)$

What is the coordinate vector $\overrightarrow{AB}$ in $\mathbb{R}^3$?

$(x_B - x_A, y_B - y_A, z_B - z_A)$

When are two vectors considered collinear?

When one is a real scalar multiple of the other.

What condition makes three vectors in R3 coplanar?

When one is a linear combination of the other two.

What forms a basis of R2?

Two non-collinear vectors.

What forms a basis of R3?

Three non-coplanar vectors.

Why do two non-collinear vectors form a basis of R2?

Because every vector has a unique linear combination of them.

What is the dot product formula for vectors in R2?

The dot product is $a_1b_1 + a_2b_2$.

What is the dot product formula for vectors in R3?

The dot product is $a_1b_1 + a_2b_2 + a_3b_3$.

When are two vectors orthogonal?

When their dot product is zero.

What is the formula for the orthogonal projection of $\vec u$ onto $\vec v$?

It is $p_{\vec v}(\vec u) = \frac{\vec u \cdot \vec v}{\|\vec v\|^2} \vec v$.

How is the dot product related to the angle $\theta$ between two vectors?

It equals $\|\vec u\| \|\vec v\| \cos(\theta)$.

What is the formula for the determinant of vectors $\vec u=(a_1,a_2)$ and $\vec v=(b_1,b_2)$?

$\det(\vec u,\vec v) = a_1b_2 - a_2b_1$

How is the area of the parallelogram spanned by two plane vectors $\vec u$ and $\vec v$ calculated?

It is the absolute value of their determinant.

What is the cross product of two non-collinear vectors in R3?

The unique vector perpendicular to both vectors.

How is the magnitude of the cross product calculated?

By $\|\vec u\wedge\vec v\|=\|\vec u\|\|\vec v\|\sin(\theta)$.

What determines the orientation of the cross product vector?

The right-hand rule.

What is the formula for the cross product of vectors $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$?

$(a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)$.

What does the norm of the cross product represent geometrically?

The area of the parallelogram spanned by the two vectors.

What is the mixed product of three vectors in R3?

The scalar triple product $\det(\vec u,\vec v,\vec w)=\vec u\cdot(\vec v\wedge\vec w)$.

What does the absolute value of the mixed product represent?

The volume of the parallelepiped spanned by three vectors.

What condition must three vectors in R3 satisfy to form a basis?

Their mixed-product determinant must be nonzero.

What is the subspace spanned by vectors $\vec u_1,\ldots,\vec u_n$?

The set of all linear combinations $\lambda_1\vec u_1+\cdots+\lambda_n\vec u_n$ with real scalars.

What happens if a vector in a generating family is a linear combination of others?

Removing it does not change the spanned subspace.

When is a vector normal to a spanned subspace?

When it is orthogonal to every vector in that subspace.

What is the vector line in R2 generated by a nonzero vector $\vec u=(a,b)$?

The set $\operatorname{Span}(\vec u)$ with direction vector $\vec u$.

What is the Cartesian equation of the vector line generated by $\vec u=(a,b)$?

$ay - bx = 0$

What is the vector plane in R3 generated by two non-collinear vectors $\vec u_1$ and $\vec u_2$?

The set $\operatorname{Span}(\vec u_1, \vec u_2)$.

What is the normal vector to the plane with Cartesian equation $ax + by + cz = 0$?

The vector $(a,b,c)$.