Recap Scalar Quantity: a quantity which can be completely described by its magnitude in a particular unit Vector Quantity: a quantity which can be completely described by its magnitude in a particular unit AND its direction Vector Notation: Geometrically: Use of a directed line segment. Vector going between Point A to
Point B: Point A initial point or tail; Point B terminal point or head Addition of two vectors: Scalar Multiplication of Vectors: Follows the triangle rule for addition. Vectors are placed head to tail. 2a is a vector parallel to a but twice
the length The negative of a vector: Subtraction of vectors: Vector of same length but point in opposite direction Let and where O is the origin Head and tail are reversed
The zero or null vector A vector which has no magnitude and no direction Algebra of Vectors 1. Closure: 2. Commutative Law: 3. Associative Law: Please note: This is the rule for the addition of three vectors placed head to tail and justified by the associative law.
4. Additive Identity Law: 5. Inverse Law: Worked Example 1: Simplifying Vector Expressions Simplify the vector expression . Collinear points Collinear: Three points which all lie on the same straight line. If and are parallel vectors then they have Point B in common and must lie on the same straight line.
A, B and C are collinear if where Worked Example 2: Collinear If , show that points A, B and C are collinear. Midpoints: Applications to Geometry: The midpoint, M, of the line segment AB where O is the origin
and A and B are points, is given by: Vectors can be used to prove many thermos of geometry. These vector proofs involve the properties of vectors discussed so far. Worked Example 3: Proof OABC is a parallelogram. P is the midpoint of OA, and the point D divides PC in the ration 1:2. Prove that O, D and B are collinear.
Hint: Draw a parallelogram first! 5.3 Vector Notation: i, j, k Connection to the Study Design AOS 4 Vectors Addition and subtraction of vectors and their multiplication by a scalar, and position vectors
Linear dependence and independence of a set of vectors and geometric interpretation Magnitude of a vector, unit vector and the orthogonal unit vectors i, j and k Key Vocabulary: Unit vector
Magnitude Circumflex Hat Cartesian coordinates Resolution of vectors Component form Linear dependence Linear combination
Linear independence Matrices Key Notation: Unit vectors Have a magnitude of 1 Unit Vectors i, j and k
Direction givers Indicated by circumflex/hat above vector Acknowledge that i, j and k are unit vectors therefore do not require
circumflex i: unit vector in the positive direction parallel to the x-axis j: unit vector in the positive direction parallel to the y-axis
k: unit vector in the positive direction parallel to the z-axis Coefficient in front of each vector represents the magnitude parallel to that particular axis Position vector Point P has coordinates relative to the origin, O
denotes position vector Vector can be expressed in terms of three other vectors: One parallel to the: x-axis y-axis z-axis
Resolution of vectors: method of splitting a vector up into its components Where Therefore; Give answers to questions in terms of i, j and k Magnitude of a Vector Magnitude of the position vector is give by: Distance between O and P is magnitude of vector.
Deduced by: Using the triangle in the xy plane, Pythagoras Theorem shows: Using triangle OPB, Worked Example 4: If the point P has coordinates , find: a) The vector b) A unit vector parallel to Addition and subtraction of vectors in three dimensions
Only add or subtract vectors in component form Add/subtract vectors i, j, k separately Vector represents the position vector of B relative to A, that is B as seen from A Worked Example 5
Two points, A and B, have the coordinates and respectively. Find a unit vector parallel to . Equality of two vectors Vectors and are equal if and only if corresponding coefficients of components are equal Scalar Multiplication of vectors
Multiply each components coefficient by scalar value Worked Example 6 If and , find the value of z if the vector is parallel to the xy plane Parallel vectors Two vectors are parallel if one is a scalar multiple of the other
That is,: is parallel to if where Worked Example 7 Given the vectors and , find the value of z in each case if: a) The length of the vector r is 8 b) The vector r is parallel to the vector s Linear dependence Linear independence
Vectors are three non-zero vectors Vectors are three non-zero vectors Vectors are said to be linearly dependent if there exist non-zero scalars such that
Vectors are said to be linearly independent if If , then we can write Only if and
This implies that one of the vectors is a linear combination of the other two Since and , it follows that and Worked Example 8 Show that the vectors , and are linearly dependent, and determine the value of z.
Direction cosines Where Generalising from the two-dimensional case: are called the direction cosines Also: Worked Example 9 Find the angle to the nearest degree that the vector makes with
the z-axis Application problems Two-dimensional case: Position vector of moving objects can be found in terms of i, j and k. i: unit vector in the east direction j: unit vector in north direction k: unit vector vertically upward Worked Example 10 Mary walks 500 metres due south, turns and
moves 400 metres due west, and then turns again to move in the direction for a further 200 metres. In all three of those movement she is at the same altitude. At this point, Mary enters a building and travels 20 metres vertically upwards in a lift. Let i, j, and k represent unit vectors of length 1 metre in the directions of east, north and vertically upwards respectively. a) Find the position vector of Mary when she leaves the lift, relative to her initial position b) Find her displacement correct to 1 decimal
place in metres from her initial point Column vector notion Vectors have similar properties to matrices Common to represent vectors as column matrices
Vectors in 3D can be represented by unit vectors Vector from the origin O to point P with coordinates can be express as: Written in matrix form:
Please note: Basic operations are performed on matrices in similar ways to those performed on vectors. Vectors and column matrices are called Isometric (Greek word meaning having the same structure) due to the reason
given above. Worked Example 11 Given the vectors represented as , and show that the points A, B and C are linearly dependent. 5.4 Scalar product and applications Connection to the Study Design AOS 4 Vectors
Scalar (dot) product of two vectors, deduction of dot product for i, j and k systems; its use to find scalar vector resolute Magnitude of a vector, unit vector and the orthogonal unit vectors i, j, k
Parallel and perpendicular vectors Key Vocabulary: Key Notation: Multiplying Vectors Think, Pair, Share When a vector is multiplied by a scalar, the resultant is a vector. Think: When multiplying two vectors together, what is the
resultant? Use drawing of vectors to aid you in your discovery. Pair: In partners, one person is A and the other is B. Share: Person A goes first to share their ideas. Person B does not talk. Only listens. Person B goes next to share their ideas. Person A does not talk. Only listens Share with the class.
Definition of the Scalar Product Scalar Product/Dot Product Read as a dot b Worked Example 12 Given the diagram below, find Properties of the Scalar Product
Resultant is a number Its commutative which means Number can be positive, negative or zero
Which follows as the angle between b and a is and The scalar product of a vector with itself is the squared magnitude of the vector. That means Scalar or common factors in a vector are
merely multiples. That is, if , then is for non zero vectors both and This implies that the sign of depends on the sign of if ; therefore is
if ; therefore is if ; therefore is Component forms i, j and k are unit vectors
The angle between vectors i and i is zero implies . Same applies for j and k Unit vectors are mutually perpendicular
If and then Worked Example 13 If and , find : Orthogonal Vectors Orthogonal means perpendicular If two vectors are orthogonal
then they at of each other Their dot product of two orthogonal vectors is zero Worked Example 14 If the two vectors and are orthogonal, find the value of z Angle between two vectors
Previously the angle made by a single vector with the x- y- or zaxis was found by using direction cosines Rearranging the Dot Product to solve for Worked Example 15 Given the vectors and , find the angle in decimal degrees between
the vectors and Finding magnitudes of vectors The magnitude and the sum or differences of vectors can be found using the properties of the scalar product Worked Example 16 If and , find
Projections Vectors can be resolved either parallel or perpendicular to the x- or y- axis Projections discuss a generalisation of the above process in which one vector is resolved parallel and perpendicular to another vector
Scalar resolute The projection of vector onto vector is defined by dropping the perpendicular from the end of a onto b (at Point C). The projection is defined as this distance along b in the direction of b.
The distance OC is called the Scalar Resolute of onto or the Scalar Resolute of parallel to Parallel vector resolute Perpendicular vector resolute
Vector resolute of onto the vector is defined as the vector along Vector resolute or component of perpendicular to the vector is is vector Vector has a length of and its direction is in the direction of the unit vector
Is obtained by subtracting the vectors: Vector resolute is given by Called the component of the vector onto the vector or parallel to vector
is obtained by: Worked Example 17 Given the vectors and , find: a) The scalar resolute of in the direction of b) The vector resolute of in the direction of c) a) The scalar resolute of perpendicular of 5.5 Vector proofs using the
scalar product Connection to the Study Design AOS 4 Vectors Vector proofs of simple geometric results, for example the diagonals of a rhombus are perpendicular, the medians of triangle are concurrent, the angle subtended by a diameter in circle is a right angle
Key Vocabulary: Key Notation: Geometrical shapes Quadrilaterals Four sided figure
No two sides are necessarily parallel nor equal in length Trapeziums Four sided figure with on pair of sides parallel but not equal Trapezium ABCD since AB is parallel to DC
Parallelograms Rectangles Four sided figure with two sets of parallel sides of equal length A parallelogram with all angles
at Parallelogram ABCD, and Rectangle ABCD,, Thus , , and hence all sides are
perpendicular Rhombuses A parallelogram with all sides equal in length Squares A Rhombus with all angles at
Triangles Median of a triangle is the line segment from a vertex to the midpoint of the opposite side Centroid of a triangle is the point of intersection of the three medians
G is the centroid of the triangle ABC and O is the origin, it can be shown that Using vectors to prove geometrical theorems Vector properties can be used to prove geometrical theorems
2. If two vectors and are parallel, then where is a scalar Following statements are useful for proving geometrical theorems 3. If two vectors and are perpendicular, then 1. O is the origin and A and B are
points, midpoint M of line segment AB given by: 4. If two vectors and are equal, then is parallel to ; furthermore, these two vectors are equal in length, so that 5. If , then the points A, B and C are collinear; that is, A, B and C all lie on a straight line Worked Example 18 Prove that if the diagonals of a
parallelogram are perpendicular, then the parallelogram is a rhombus. Parametric equations Connection to the Study Design AOS 4 Vectors Key Vocabulary: Key Notation:
Parametric equations Worked Example 19 Given the vector equation , find and sketch the Cartesian equation of the path, and state the domain and range Eliminating the parameter Worked Example 20 Given the vector equation , find
and sketch the Cartesian equation of the path, and state the domain and range Worked Example 21 Given the vector equation , find and sketch the Cartesian equation of the path, and state the domain and range Parametric representation Worked Example 22
Show that the parametric equations and where represent the hyperbola Sketching parametric curves
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