What is a Vector?

Vector: A mathematical object possessing both magnitude (size) and direction, describing a translation in space. Vectors are distinct from scalars, which have magnitude only.

Vectors can be written in two common notations:
$v OR v$

Because a vector encodes both how far and which way, two arrows drawn anywhere in space that have the same length and point in the same direction represent the identical vector. This is the free vector concept , position of the arrow does not matter, only its magnitude and direction.

Analogy

Think of a vector like a walking instruction: "Walk 5 km due North." It doesn't matter where you start , the instruction is the same. A scalar would just say "Walk 5 km" with no direction given.

Warning

Do not confuse vectors with points or lines. A vector describes a displacement, not a fixed location. Only a position vector (rooted at the origin) pins a vector to a specific point in the coordinate system.

Position Vectors and Displacement Vectors

Position Vector: The vector $O P$ that runs from the fixed origin $O$ to a point $P$ . It uniquely describes the location of $P$ in the coordinate system.

Displacement Vector: The vector $A B$ that describes the directed change in position from point $A$ to point $B$ , regardless of the origin.

If point $A$ has position vector $a = O A$ and point $B$ has position vector $b = O B$ , then the displacement vector from $A$ to $B$ is:
$A B = b - a$

Notice the direction matters: $B A = a - b = - A B$ .

Note

A position vector is technically also a displacement vector , it just happens to start at the origin. That is why we write $A B = b - a$ using the position vectors of the two endpoints.

Example

Let $A = (1, 3, 4)$ so $a = 134$ , and $B = (3, 2, 5)$ so $b = 325$ .

Then:
$A B = b - a = 3 - 1 2 - 3 5 - 4 = 2 - 1 1$
$B A = a - b = - 2 1 - 1$

Note that $B A = - A B$ , confirming the vectors are equal in magnitude but opposite in direction.

Position Vectors and Displacement Vectors

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9 more sections in this topic

What is a Vector?

Vector: A mathematical object possessing both magnitude (size) and direction, describing a translation in space. Vectors are distinct from scalars, which have magnitude only.

Vectors can be written in two common notations:
$v OR v$

Analogy

Think of a vector like a walking instruction: "Walk 5 km due North." It doesn't matter where you start , the instruction is the same. A scalar would just say "Walk 5 km" with no direction given.

Warning

Position Vectors and Displacement Vectors

Position Vector: The vector $O P$ that runs from the fixed origin $O$ to a point $P$ . It uniquely describes the location of $P$ in the coordinate system.

Displacement Vector: The vector $A B$ that describes the directed change in position from point $A$ to point $B$ , regardless of the origin.

If point $A$ has position vector $a = O A$ and point $B$ has position vector $b = O B$ , then the displacement vector from $A$ to $B$ is:
$A B = b - a$

Notice the direction matters: $B A = a - b = - A B$ .

Note

A position vector is technically also a displacement vector , it just happens to start at the origin. That is why we write $A B = b - a$ using the position vectors of the two endpoints.

Example

Let $A = (1, 3, 4)$ so $a = 134$ , and $B = (3, 2, 5)$ so $b = 325$ .

Then:
$A B = b - a = 3 - 1 2 - 3 5 - 4 = 2 - 1 1$
$B A = a - b = - 2 1 - 1$

Note that $B A = - A B$ , confirming the vectors are equal in magnitude but opposite in direction.

Free preview

AHL 3.12—Vector definitions

What is a Vector?

Position Vectors and Displacement Vectors

9 more sections in this topic

AHL 3.12—Vector definitions

What is a Vector?

Position Vectors and Displacement Vectors

9 more sections in this topic