Vector definitions (HL)

Displacement vectors can be translated in the plane or space, while maintaining their magnitudes and directions. Meanwhile position vectors represent the vector from origin to a specific point in space.

Geometrically, they are represented by arrows.

Definitions

A vector or vector quantity, $\bm{v}$ or $\vec{v}$, is associated with a magnitude (size) and a direction. Magnitudes are non-negative, and are represented by $\lvert\bm v\rvert$, $\lvert\vec v\rvert$, or simply $v$

In contrast, a scalar or scalar quantity, $v$, is only associated with a magnitude. Though, a scalar could be negative.

Scalar multiplication, multiplying a vector (or scalar) by a scalar. This multiplies the vector’s magnitude by the scalar. If the scalar is positive, the resulting direction is same as before; if it is negative, the vector reverses direction.

The zero vector, $\bm{0}$, or $\vec{0}$, is the vector with zero magnitude and in any arbitrary direction.

The unit vector, $\bm{\hat{v}}$, is the vector in the direction of $\bm{v}$ and with magnitude $1$.

$\bm i$, $\bm j$, $\bm k$ are the unit vectors for the $x$, $y$, and $z$-axes. The axes are perpendicular to each other and intersect at the origin $(0, 0, 0)$

A position vector is a vector from the origin to a specific point in space. Eg point $A(1, -2, 3)$ is associated with the vector $\overrightarrow{OA} = (1, -2, 3)$. Position vectors are fixed in space. For the rest of the vectors discussion, position vectors are synonymous with “points”.

A displacement vector is a difference between two position vectors, or between two displacement vectors. For instance, $\overrightarrow{AB} = (2, -1, 0)$ is the displacement from $\overrightarrow{OA} = (-3, 2, 4)$ to $\overrightarrow{OB} = (-1, 1, 4)$.

Displacement vectors are more powerful and useful because they can be moved around. Common types of displacement vectors include

• a displacement vector representing a line segment
• a direction vector of a line
• a direction vector on a plane
• a normal vector to a plane

In certain cases, such as when the displacement vector is used as a direction vector of a line or a normal vector of a plane, only the direction is of interest.

A vector goes from its tail to its head. For example, the tail of $\overrightarrow{OP}$ is $O$, while the head is $P$. Position vectors have fixed tails and heads, while displacement vectors have tails and heads that can be translated together.

A vector is said to be the vector sum of its $x$, $y$, and $z$ components. This is true for both position and displacement vectors. Namely

The components are linearly independent, or that they are perpendicular to each other.