Use of Vectors in i, j and k unit

Use of Vectors in i, j and k unit

Unit Vectors i, j and k

• Unit vectors are vectors with a magnitude (or length) of 1.
• i, j and k are the standard unit vectors in three-dimensional space (3D).
• i is a unit vector in the x-axis direction, j is a unit vector in the y-axis direction and k is a unit vector in the z-axis direction.

Representation of Vector in i, j, k Form

• A vector in a 3D space can be represented in i, j, and k form: ai + bj + ck, where a, b, and c are its components in the x, y, and z directions respectively.
• The vector OP with initial point O(0,0,0) and terminal point P(x,y,z) can be written as x * i + y * j + z * k.

Arithmetic Operations with Unit Vectors i, j, k

• Addition and subtraction of vectors in i, j, and k forms involves simply adding or subtracting their corresponding components.
• The scalar multiplication of a vector in i, j, k form involves distributing the scalar to each component.
• For example, 2*(ai + bj + ck) = 2ai + 2bj + 2ck.

Dot Product in i, j, k Form

• The dot product of two vectors in i, j, k form, e.g. (ai + bj + ck).(di + ej + fk), is computed as ad + be + cf.
• This results in a scalar quantity.

Cross Product in i, j, k Form

• The cross product of two vectors in i, j, k form, is a vector orthogonal to the original vectors.
• Given v = ai + bj + ck and w = di + ej + fk, the cross product v x w is (bf - ce)i - (af - cd)j + (ae - bd)k.

Rules for Manipulation using i, j, k Form

• Operations involving vectors in i, j, k form obey the same commutative, associative, and distributive properties as standard algebraic operations.
• However, it’s important to note that the cross product is not commutative: v x w != w x v. In fact, v x w = - (w x v).