For AI agents: use /llms.txt for the Nakafa content index.
A three-dimensional vector is a quantity that has both magnitude and direction in three-dimensional space. Unlike two-dimensional vectors that exist only on a plane (x x x y y y x x x y y y
Three-dimensional vectors can be notated in various ways:
Letter notation with an arrow above it: a ⃗ \vec{a} a P Q → \overrightarrow{PQ} P Q
Component notation: ( a x , a y , a z ) (a_x, a_y, a_z) ( a x , a y , a z ) ( a 1 , a 2 , a 3 ) (a_1, a_2, a_3) ( a 1 , a 2 , a 3 )
Basis notation: a x i ⃗ + a y j ⃗ + a z k ⃗ a_x\vec{i} + a_y\vec{j} + a_z\vec{k} a x i + a y j + a z k
A vector in three-dimensional space consists of three components that represent the projection of the vector on each coordinate axis:
a x a_x a x x x x a y a_y a y y y y a z a_z a z z z z i ⃗ , j ⃗ , k ⃗ \vec{i}, \vec{j}, \vec{k} i , j , k The magnitude or length of a three-dimensional vector a ⃗ = ( a x , a y , a z ) \vec{a} = (a_x, a_y, a_z) a = ( a x , a y , a z )
If a ⃗ = ( 3 , 4 , 5 ) \vec{a} = (3, 4, 5) a = ( 3 , 4 , 5 ) a ⃗ \vec{a} a
Addition and subtraction of three-dimensional vectors are performed by adding or subtracting the corresponding components.
If a ⃗ = ( a x , a y , a z ) \vec{a} = (a_x, a_y, a_z) a = ( a x , a y , a z ) b ⃗ = ( b x , b y , b z ) \vec{b} = (b_x, b_y, b_z) b = ( b x , b y , b z )
Multiplying a scalar k k k a ⃗ \vec{a} a k > 0 k > 0 k > 0 k < 0 k < 0 k < 0 ∣ k ∣ times |k|\text{ times} ∣ k ∣ times a ⃗ \vec{a} a
The dot product between two vectors a ⃗ \vec{a} a b ⃗ \vec{b} b
where θ \theta θ
The dot product has the following properties:
a ⃗ ⋅ b ⃗ = b ⃗ ⋅ a ⃗ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} a ⋅ b = b ⋅ a a ⃗ ⋅ b ⃗ = 0 \vec{a} \cdot \vec{b} = 0 a ⋅ b = 0
a ⃗ ⋅ a ⃗ = ∣ a ⃗ ∣ 2 \vec{a} \cdot \vec{a} = |\vec{a}|^2 a ⋅ a = ∣ a ∣ 2
The cross product between two vectors a ⃗ \vec{a} a b ⃗ \vec{b} b c ⃗ \vec{c} c
The magnitude of the cross product is:
where θ \theta θ
Three-dimensional vectors have many applications in various fields:
Physics : To represent force, velocity, acceleration, and momentum in three-dimensional spaceComputer Graphics : To represent position and movement of objects in three-dimensional spaceRobotics : To control robot movement in spaceNavigation : To determine direction and distance in three-dimensional spaceMechanical Engineering : For structural analysis and fluid mechanics
