Vectors

What you need to know How to add and subtract vectors and multiply by a scalar How to find the magnitude of a vector How to calculate the scalar product and how it relates to angles between vectors How to use vectors for co-ordinate geometry How to use the vector eqn of lines in the form r = a +tb How to find the intersection of lines, or show that they are skew lines (non-parallel and non-intersecting) How to find the angle between lines

The Length of a 3D Vector Find the length of the vector 3i + 5j - 2k = √ 3 2 + 5 2 + ( -2 ) 2 = √ 38

The vector equation of a straight line passing through the point A with position vector a, and parallel to the vector u R u R = a general point on the line A OR = OA + AR r r = a + λu a O where λ is a scalar parameter

The vector equation of a straight line passing through the points A and B, with position vectors a and b. R AB = b - a R = a general point on the line B OR = OA + AR A OR = OA + λAB b a r = a + λ( b – a ) O where λ is a scalar parameter

Vector Equation of a Line in 3D Example 1 Find the equation of the line passing through the point ( 3, 4, 1 ) and ( 5, 7, 7 ) Equation is given by r = a + λb r = 2 3 6 ( ) 4 1 + λ

The Angle Between Two Vectors θ b scalar product ( dot product ) cos θ = a . b |a| |b| Note: If two vectors a and b are perpendicular then a . b = 0

The Angle Between Two Directions Example 2 Find the angle between the following pair of vectors: a = i + 4j + 2k and b = 2i + 5j + 7k |a| = √ ( 1 2 + 4 2 + 2 2) = √ 21 |b| = √ ( 2 2 + 5 2 + 7 2 )= √ 78 a . b = 1 x 2 + 4 x 5 + 2 x 7 = 36 cos θ = a . b |a| |b| cos θ = 36 √21 √78 θ = 27.2°

The Angle Between Two Lines in 3D Example 3 r = 2 -1 ( ) 1 4 + λ r = 3 1 ( ) 2 -1 + μ The angle between the lines is the angle between their directions cos θ = a . b |a| |b| cos θ = 2 x 3 + (-1) x 0 + (-1) x 1 √2 2 + (-1) 2 + (-1) 2 √3 2 + 0 2 + 1 2 cos θ = 5 √ 6 √ 10 θ = 49.8°

4. Past paper question Jan09 no7 Show that the straight line with equation meets the line passing through (9, 7, 5) and (7, 8, 2) and find the point of intersection. 2nd line eqn is Intersect when

4. Past paper question Jan09 no7 So lines intersect at (5, 9, -1)

Past paper question Jan08 no5  Example 5 The vector equations of two lines are and Prove that the two lines are perpendicular and skew

Past paper question Jan08 no5  Use the direction vectors only and Lines are perpendicular when dot product =0

Past paper question Jan08 no5  would cross when Solving top two eqns gives

Past paper question Jan08 no5  Substituting into equations gives So lines do not intersect They are not parallel (seen in part (i)) So they are skew lines You have to have both bits here

Additional example C is the point (3,12,3) and the point P is on the line CP is perpendicular to the line. Find the co-ordinates of P

CP is perpendicular to the line – We need scalar product with the direction vector