ESAT Maths 1 · free worked solution

ESAT Maths 1 vectors: calculate a magnitude

Vectors solution: combine components, then use Pythagoras to find the magnitude √101.

Vector magnitudeSpecification M5.19Reviewed 9 August 2026

Answer first

Option A

The correct answer is

101\sqrt{101}

Do the vector arithmetic component by component before applying the magnitude formula.

This is original UniGenius practice material from the reviewed Maths 1 bank. It is not an official UAT-UK question, past paper, or recalled question.

The question

Read the setup, then choose the shortest valid route

The vectors u=(3,−4)u=(3,-4) and v=(5,2)v=(5,2) are given. What is the magnitude of the vector 2u−v2u-v?

  • A101\sqrt{101}Answer
  • B98\sqrt{98}
  • C99\sqrt{99}
  • D1010
  • E110\sqrt{110}

Worked solution

The route in 4 steps

  1. Multiply the first vector by the scalar: 2u=(6,−8)2u=(6,-8).

  2. Subtract the second vector componentwise: 2u−v=(6,−8)−(5,2)=(1,−10)2u-v=(6,-8)-(5,2)=(1,-10).

  3. The squared magnitude is 12+(−10)2=1011^2+(-10)^2=101.

  4. Therefore the magnitude of 2u−v2u-v is 101\sqrt{101}.

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