Vectors A vector quantity has magnitude and direction. Vectors can be denoted by AB, a, AB (with an arrow above, or a line below, the letters) If a = then a column vector looks like:
Multiplication of a vector by scalar. Eg a = , then 2a =
Add two vectors, add numbers in the same positions. … this is similar to horizontal and vertical components. The magnitude or modulus of a vector is |a| and can be calculated by Pythagoras's theorem. If x = and y = , The sum(resultant) is .ie √(-32 + 42) = √(25) = 5.
Vectors can be added and subtracted using scaled diagrams. ex Determime a-b. Unit Vectors A unit vector indicates direction and has a magnitude of 1. The unit vector in the direction of the x-axis is i and the unit vector in the direction of the y-axis is j. On a graph, 3i + 4j would be at (3 , 4). This is another method of specifying vectors. It also makes adding and subtracting vectors easy: you just add the i terms together and add the j terms together.
For example: (3i + j) + (5i - 4j) = 8i - 3j. This is the same as writing it as:
Letters used to represent vectors should always be underlined or in bold type. eg, the velocity of an object may be represented by bold type lower case v.. There are three important and common unit vectors in the directions of the x, y and z-axes. The unit vector in the direction of the x-axis is i, the unit vector in the direction of the y-axis is j and the unit vector in the direction of the z-axis is k. the vector 5i - 3j would look like this on a diagram: Adding Vectors If the vectors are given in unit vector form, then add together the i, j and k values. Example p = 3i + j, q = -5i + j. Find p + q. Since the vectors are given in i, j form, we can easily calculate the resultant. 3i + j - 5i + j = -2i + 2j . This could also have been worked out from a diagram:
ex Hence determine p-q
The Magnitude of a Vector The magnitude of a vector can be found using Pythagoras's theorem. The magnitude of ai + bj = √(a2 + b2)
Resolving a Vector Resolving a vector means finding its magnitude in a particular direction.
In the diagram above, the vector r has magnitude r and direction Ø to the x-axis. Using basic trigonometry, the component of r in the direction of the x-axis is rcos Ø. The component in the direction of the y-axis is rsinØ. Therefore r = rcos Øi + rsinØj.
Examples (a) Determine the following by calculation and scale drawing (1) if x = 3i -2j and y = 6j – 5i, determine resultant of x +y and x-y
x +y = 4j -2i = 4.47 /116.6o and x-y = 8i – 8j = 11.3 / -45o
(2) If x = 2i+2j+2k and y = 2i-2j+2k, determine resultant of x+y and x-y.
x + y = 4i +0+ 4k, = 5.66 /45o horizontal plane, x-y = 0+4j +0 = 4/90o Vertical plane
(3) Calculate the resultant of x+y+z,: x=3i+2j-5k, y=2i-2j-2k, z= -4i-5j-6k
x+y+z = 1i -5j -13k = 13.96, / -4.4o horizontal, /-11.3o vertical.
(4)An aircraft with a nominal air speed of 700 km/h is due to fly 1000km due North. There is a cross wind from east to west of 100km/h which blows the aircraft off course. (a) In what direction is the plane blown and what is its resultant velocity ?. (b) In what direction must the plane fly in to reach its destination (c) How long will the plane take to reach its destination?
(a) Plane is blown West of North. Resultant velocity = 692.8 k/hr
(b) Direction is 8.21o East of North
© Time = 1000 / 692.8 = 1.443hours = 1hr 26mins 35secs
(5) A projectile is fired, due East at an angle of 76o to the ground. Determine the values of the 2 dimensional horizontal and vertical components x and y.
x = Vcos76o y = Vsin76o
(6) A yacht sails 2.5km south-west, then 3.0km north-west and finally a further distance in an unknown direction arriving 5km due north of its starting position. Find the direction and magnitude of the third leg of its voyage?
Magnitude = 6.06km. Direction 39.96o East of North.