Magnitude of a vector revision notes

Edexcel International GCSE Maths (4MA1), chapter 27: Vectors, specification point 5.1e. Free lesson.

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The magnitude, or modulus, of a vector is the length of its arrow. You find it using Pythagoras on the two numbers in the column vector, then scale that length when the vector is multiplied by a number.

How to find a vector's length

A column vector gives the horizontal and vertical sides of a right-angled triangle. The vector itself is the sloping side, so use Pythagoras to find its magnitude. The formula is not always printed in an exam, so learn it. Modulus and magnitude both mean the length of the vector. For a = (x, y), |a| = √(x² + y²). The top number, x, tells you how far across the vector goes. The bottom number, y, tells you how far up it goes.

How negative numbers affect the length

A negative component shows direction, such as going down or left. It does not make the length negative, because squaring removes the minus sign. A vector's magnitude is always a positive length. For a = (5, −12), the triangle has sides 5 and 12. |a| = √(5² + (−12)²) = √(25 + 144) = √169 = 13. The vector is the sloping side, not the two shorter sides added together.

Why you do not add the two numbers

Adding the components gives the distance along two sides of the triangle, not the straight-line length of the vector. The vector cuts across the corner, so its length is found by squaring, adding and rooting. For the sides 5 and 12, 5 + 12 = 17 is not the vector's magnitude. Pythagoras gives √(5² + 12²) = 13, the straight-line length. A length is never negative, and it is not simply x + y.

How multiplying a vector changes its length

Multiplying a vector by a scalar (a number that scales a vector) changes its length by the scalar's size. A negative scalar reverses the direction but leaves the length unchanged apart from that scaling. For b = (−4, 3) cm, |b| = √((−4)² + 3²) = √25 = 5 cm. Therefore |3b| = 3|b| = 3 × 5 = 15 cm. You do not need to draw a new triangle once you know |b|.

Check a magnitude

Use the same method for any column vector, including one with a negative component. Square both numbers before adding them. For c = (−6, 8), |c| = √((−6)² + 8²) = √(36 + 64) = √100 = 10.