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Uses of the Scalar Product (Cambridge (CIE) A Level Maths): Revision Note

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Uses of the Scalar Product

This revision note covers several applications of the scalar product for vectors – namely, how you can use the scalar product to:

  • find the angle between vectors or lines

  • test whether vectors or lines are perpendicular

  • find the closest distance from a point to a line

How do I find the angle between two vectors?

 

  • Recall that a formula for the scalar (or ‘dot’) between vectors bold a and bold b is

bold a bold times bold b equals open vertical bar bold a close vertical bar open vertical bar bold b close vertical bar cos invisible function application theta

  • where theta is the angle between the vectors when they are placed ‘base to base’

    • that is, when the vectors are positioned so that they start at the same point

  • We arrange this formula to make cos space theta the subject:

  • To find the angle between two vectors

    • Calculate the scalar product between them

    • Calculate the magnitude of each vector

    • Use the formula to find cos space theta

    • Use inverse trig to find theta

How do I find the angle between two lines?

  • To find the angle between two lines, find the angle between their direction vectors

  •  For example, if the lines have equations bold r equals bold a subscript 1 plus s bold d subscript 1 and bold r equals bold a subscript 2 plus t bold d subscript 2, then the angle theta between the lines is given by

theta equals cos to the power of negative 1 end exponent open parentheses fraction numerator bold d subscript 1 bold times bold d subscript 2 over denominator open vertical bar bold d subscript 1 close vertical bar open vertical bar bold d subscript 2 close vertical bar end fraction close parentheses

 

How do I tell if vectors or lines are perpendicular?

  • Two (non-zero) vectors bold a and bold b are perpendicular if, and only if, bold a bold times bold b equals 0

    • If the a and b are perpendicular then:

      • theta equals 90 degree rightwards double arrow cos space theta equals 0 rightwards double arrow open vertical bar bold a close vertical bar open vertical bar bold b close vertical bar cos space theta blank equals 0 rightwards double arrow bold a bold times bold b equals 0

    • If  bold a bold times bold b equals 0 then:

      • open vertical bar bold a close vertical bar open vertical bar bold b close vertical bar cos space theta blank equals 0 rightwards double arrow cos space theta equals 0 rightwards double arrow theta equals 90 degree rightwards double arrow a and b are perpendicular

    • For example, the vectors 2 i minus 3 j plus 5 k and negative 4 i minus j plus k  are perpendicular since

open parentheses 2 i minus 3 j plus 5 k blank close parentheses times open parentheses negative 4 i minus j plus k close parentheses equals 2 cross times open parentheses negative 4 close parentheses plus open parentheses negative 3 close parentheses cross times open parentheses negative 1 close parentheses plus 5 cross times 1 equals negative 8 plus 3 plus 5 equals 0

How do I find the shortest distance from a point to a line?

  • Suppose that we have a line l with equation bold r equals bold a plus t bold d  and a point P not on l

  • Let F be the point on l which is closest to bold italic P (sometimes called the foot of the perpendicular)

    • Then the line between F and P will be perpendicular to the line l

  • To find the closest point F

    • Call bold f equals OF with rightwards arrow on top and bold p equals stack O P with rightwards arrow on top

    • Since F lies on l, we have bold f equals bold a plus t subscript 0 bold d, for a unique real number t subscript 0

    • Find the vector stack bold italic F bold italic P with rightwards arrow on top using bold p minus bold f

    • stack bold italic F bold italic P with rightwards arrow on top is perpendicular to bold italic d so form an equation using open parentheses bold p minus bold f close parentheses times bold d equals 0

    • Solve this equation to find the value of t subscript 0

    • Use the value of t subscript 0 to find bold f

  • The shortest distance between the point  and the line  is the length

  • Note that the shortest distance between the point and the line is sometimes referred to as the length of the perpendicular

7-3-4-foot-of-the-perpendicular

Worked Example

7-3-4-uses-of-scalar-product-we-solution-part-1
7-3-4-uses-of-scalar-product-we-solution-part-2

Examiner Tips and Tricks

It can be easier and clearer to work with column vectors when dealing with scalar products.

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