Components of Vector in Algebra

A vector in three-dimensional geometry can be written in terms of its components along the x-, y-, and z-axes using the unit vectors \[\vec{i}, \vec{j}, \vec{k}\]. This component form makes it easier to find magnitude, compare vectors, and perform operations like addition and subtraction.

If P(x, y, z) is a point, then its position vector is

This is called the component form of a vector.

• Magnitude
  \[|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\]

• Addition
  If \[\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\] and
  \[\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\]
  then \[\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}\]

• Subtraction
  \[\vec{a} - \vec{b} = (a_1 - b_1)\hat{i} + (a_2 - b_2)\hat{j} + (a_3 - b_3)\hat{k}\]

• Scalar Multiplication
  
  \[\lambda\vec{a} = (\lambda a_1)\hat{i} + (\lambda a_2)\hat{j} + (\lambda a_3)\hat{k}\]

• Equality of Vectors
  Two vectors are equal if their corresponding components are equal.

• Collinearity of Vectors
  

  Two vectors are collinear if one is a scalar multiple of the other, i.e. \[\vec{b} = \lambda\vec{a}\]

• Equality of Vectors
  Two vectors are equal if their corresponding components are equal.

Find the values of \[x, y\] and \[z\] so that the vectors \[\vec{a} = x\hat{i} + 2\hat{j} + z\hat{k}\]and \[\vec{b} = 2\hat{i} + y\hat{j} + \hat{k}\]are equal.

Solution: Note that two vectors are equal if and only if their corresponding components are equal. Thus, the given vectors \[\vec{a}\] and \[\vec{b}\] will be equal if and only if

Question: Find a vector of magnitude 7 units in the direction of:

Solution:

Step 1: Find the magnitude of \[\vec{a}\].

Step 2: Find the unit vector in the same direction.

Step 3: Multiply by 7.