# How to work out the magnitude (modulus) of a vector in the form xi + yj + zk.

To work out the magnitude (or modulus) of a vector in the form x**i** + y**j** +z**k **all you need to do is square the coefficients of** i**, **j** and **k**, add these squares together and square root the answer:

√(x²+y²+z²)

**Example 1**

If **a** = 3**i** + 5**j** – 3**k**, work out |**a**|.

|**a**| stands for the magnitude of the vector, so substitute x = 3, y = 5 and z = -3 into the above formula:

|**a**| = √(x²+y²+z²)

= √(3² + 5² + (-3)²)

= √(9 + 25 + 9)

= √43

**Example 2**

If **b** = 2**i** + 9**j** – 4**k**, work out |**b**|.

|**b**| stands for the magnitude of the vector, so substitute x = 2, y = 9 and z = -4 into the above formula:

|**b**| = √(x²+y²+z²)

= √(2² + 9² + (-4)²)

= √(4 + 81 + 16)

= √101

**Example 3**

If **c** = 10**i** + 4**j** – 7**k**, work out |**c**|.

|**c**| stands for the magnitude of the vector, so substitute x = 10, y = 4 and z = -7 into the above formula:

|**c**| = √(x²+y²+z²)

= √(10² + 4² + (-7)²)

= √(100 + 16 + 49)

= √165

**Example 4**

If **a** = 3**i** – 5**j** – 7**k** and **b** = 2**i** +5**j** –**k** work out |**AB**|. Round your answer off to 1 decimal place.

Before you can work out the magnitude you will need to work out the vector **AB**:

To find **AB** work out **b** –**a**:

(3**i** -5**j** -7**k**) – (2**i** +5**j**-**k**) =** i** – 10**j** -6**k**

So **AB** = **i** – 10**j** -6**k**

All you need to do next is work out the magnitude of **AB**. The coefficient of x,y and z are 1, 10 and -6:

|**AB**| = √(1² + 10² + (-6)²) = √137

= 11.7 to 1 decimal place.

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