# How to Find Magnitude of Vectors?

A vector is an object that has both magnitude and direction. To find the magnitude of a vector, we need to calculate the length of the vector. Let's talk more about the magnitude of a vector in this post.

A vector has a direction and a magnitude. The magnitude of a vector refers to the size or length of the vector.

## Step by step guide to magnitude of vectors

• The length of a vector determines its magnitude. The magnitude of the vector a is represented by the letter $$∥a∥$$. The magnitude of a vector is always a positive or zero number, meaning it cannot be a negative number.
• For a two-dimensional vector a$$=(a_1,a_2)$$, the formula for its magnitude is:

$$\color{blue}{ ∥a∥}$$ $$\color{blue}{ =\sqrt{a_1^2+a_2^2}}$$

• For a three-dimensional vector a$$=(a_1,a_2,a_3)$$,the formula for its magnitude is:

$$\color{blue}{ ∥a∥}$$ $$\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2}}$$

• The formula for a vector’s magnitude can be applied to any set of dimensions. For example, if a$$=(a_1,a_2,a_3,a_4)$$, is a four-dimensional vector, the magnitude formula is:

$$\color{blue}{ ∥a∥}$$ $$\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2+a_4^2}}$$

### Magnitude of Vectors – Example 1:

Find the magnitude of the vector with $$𝑣⃗=(3,5)$$.

Use this formula to find the magnitude of the vector: $$\color{blue}{ ∥a∥}$$ $$\color{blue}{ =\sqrt{a_1^2+a_2^2}}$$

$$∥ 𝑣⃗∥$$$$=\sqrt{3^2+5^2}$$

$$=\sqrt{9+25}$$$$=\sqrt{34}$$

$$∥ 𝑣⃗∥$$ $$=\sqrt{34}≅5.83$$

### Magnitude of Vectors – Example 2:

Find the magnitude of the vector $$𝑣⃗ =3i+4j−5k$$.

Use this formula to find the magnitude of the vector: $$\color{blue}{ ∥a∥}$$ $$\color{blue}{ =\sqrt{a_1^2+a_2^2+a_3^2}}$$

$$∥ 𝑣⃗∥$$ $$=\sqrt{(3^2)+(4^2)+(-5^2)}$$

$$=\sqrt{(9+16+25)}=\sqrt{50}$$

$$∥ 𝑣⃗∥$$ $$=\sqrt{50}=5\sqrt{2}$$

## Exercises for Magnitude of Vectors

### Find the magnitude of the vector.

1. $$\color{blue}{𝑣⃗ =5i-4k+3j}$$
2. $$\color{blue}{𝑣⃗ =6,8}$$
3. $$\color{blue}{𝑣⃗ =4i-3k+2j-5l}$$
4. $$\color{blue}{𝑣⃗ =4,-3}$$
1. $$\color{blue}{∥ 𝑣⃗∥}$$$$\color{blue}{=5\sqrt{2}}$$
2. $$\color{blue}{∥ 𝑣⃗∥}$$$$\color{blue}{=10}$$
3. $$\color{blue}{∥ 𝑣⃗∥}$$$$\color{blue}{=3\sqrt{6}}$$
4. $$\color{blue}{∥ 𝑣⃗∥}$$$$\color{blue}{=5}$$

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