VECTORS IN COMPONENT FORM Vector as position vector

VECTORS IN COMPONENT FORM Vector as position vector of point A in 3 – D in Cartesian coordinate system: Both, position vector of point A and point A have the same coordinates:

VECTOR BETWEEN TWO POINTS

Unit vector Definition A unit vector is a vector whose length is 1. It gives direction only!

PARALLEL and COLLINEAR VECTORS

ARE 3 POINTS COLLINEAR ? How can you check it: 1. Form two vectors with these three points. They will definitely have one common point. 2. Check if these two vectors are parallel. Show that P(0, 2, 4), Q(10, 0, 0) and R(5, 1, 2) are collinear.

THE DIVISION OF A LINE SEGMENT A = (2, 7, 8) B = ( 2, 3, 12) INTERNAL DIVISION P divides [AB] internally in ratio 1: 3. Find P point P is (2, 6, 9) EXTERNAL DIVISION X divide [AB] externally in ratio 2: 1, or X divide [AB] in ratio – 2: 1. Find Q point Q is (2, – 1, 16)

DOT/SCALAR PRODUCT Definition θ or: Product of the length of one of them and projection of the other one on the first one

In Cartesian coordinates:

Properties of dot product

CROSS / VECTOR PRODUCT Definition

In Cartesian coordinates: Using properties of determinates We can write cross product in simple form:

Properties of vector/cross product

How do we use dot and cross product • To find angle between vectors the easiest way is to use dot product, not vector product. • Angle between vectors can be acute or obtuse • Angle between lines is by definition acute angle between them, so Dot product of perpendicular vectors is zero. • To show that two lines are perpendicular use the dot product with line direction vectors. • To show that two planes are perpendicular use the dot product on their normal vectors.

Volume of a parallelepiped = scalar triple product TEST FOR FOUR COPLANAR POINTS

Are the points A(1, 2, -4), B(3, 2, 0), C(2, 5, 1) and D(5, -3, -1) coplanar?

A line is completely determined by a fixed point and its direction. Using vectors gives us a very neat way of writing down an equation which gives the position vector of any point P on a given straight line. This method works equally well in two or three dimensions.

LINE EQUATION IN 2 – D and 3 – D COORDINATE SYSTEM ● Vector equation of a line ● Cartesian equation of a line

Find the equation of the line passing through the points A(3, 5, 2) and B(2, -4, 5). Find the direction of the line: One possible direction vector is The Cartesian equation of this line is (using the coordinates f point A). The equivalent vector equation is

ANGLE BETWEEN TWO LINES Two vectors

Shortest distance from a point to a line

Relationship between lines 2 – D: 3 – D: ● the lines are coplanar (they lie in the same plane). They could be: ▪ intersecting ▪ parallel ▪ coincident ● the lines are not coplanar and are therefore skew (neither parallel nor intersecting)

Distance between two skew lines (sometimes I see it, sometimes I don’t)

PLANE EQUATION A plane is completely determined by two intersecting lines, what can be translated into a fixed point A and two nonparallel direction vectors ● Normal/Scalar product form of vector equation of a plane ● Cartesian equation of a plane

What does the equation 3 x + 4 y = 12 give in 2 and 3 dimensions? https: //www. osc-ib. com/ib-videos/default. asp http: //www. globaljaya. net/secondary/IB/Subjects%20 Report/May%202012%20 s ubject%20 report/Maths%20 HL%20 subject%20 report%202012%20 TZ 1. pdf

Find the equation of the plane passing through the three points P 1(1, -1, 4), P 2(2, 7, -1), and P 3(5, 0, -1). Find the equation of the plane with normal vector containing point (-2, 3, 4). vector form: Find the distance of the plane from the origin, and the unit vector perpendicular to the plane.

ANGLES ● The angle between a line and a plane ● The angle between two planes is the same as the angle between their 2 normal vectors take acute angle

● INTERSECTION OF TWO or MORE PLANES