UNIT 8: Applications of Vectors
7.3 - The Dot Product of Two Geometric Vectors
Properties of Dot Product
Associative property with a VECTOR is NOT TRUE
Example
Right Side
Left Side
PROPERTY 5:
PROPERTY 4:
Dot Product
The dot product of two vectors is a SCALAR QUANTITY. (Also referred to as the scalar product)
The VALUE of the dot product is determined by the angle between the two vectors.
When the angle between the two vectors is OBTUSE, the dot product is LESS than 0.
When the angle between the two vectors is 90 degrees, meaning they are PERPENDICULAR, the dot product is EQUAL to 0.
When the angle between the two vectors is ACUTE, the dot product is GREATER than 0.
7.2 - Velocity
Applications
Example: The resultant ground speed and course of the plane in this example could be calculated through the following procedure:
STEP 1:
STEP 2:
DIRECTION (There are 3 ways to refer to the direction of the resultant vector) -
3. Using the locations of the given vectors as general direction indicators.
Example of Option 3: Indicates that the angle between the vector 4N and the resultant vector is 54.79 degrees (in the direction of vector 6N).
2. Using the true bearings - the angle from due north to the vector (in a clockwise motion).
Example of Option 2: Indicates that the vector is pointing (the direction of the head) towards 54.79 degrees right from due north.
1. Using conventional bearings - the locations of North, East, South, and West as general direction indicators, and then specifying the exact angle that separates the vector from them.
Example of Option 1: Indicates that the vector is pointing (the direction of the head) towards 54.79 degrees east of north.
The resultant (ground speed) in velocity applications is the speed of the plane/boat RELATIVE to an individual on the GROUND, INCLUDING the effects of wind/current on the speed of the air/water.
Velocity
Magnitude of velocity is speed.
VELOCITY is a vector quantity (both direction and magnitude are important).
7.1 - Vectors as Forces
Resolving a Vector
1. A right triangle is created with the given vector.
2. The magnitudes of the components can then be determined through using primary trigonometric ratios and a given angle.
In order to RESOLVE a vector, a single force must be decomposed into its horizontal and vertical components.
Equilibrium
In order to maintain equilibrium with three forces on a plane, the forces must be able to create a triangle, which occurs when the triangle inequality holds TRUE.
TRIANGLE INEQUALITY: The sum of any two sides is greater than or equal to the third side.
EQUILIBRANT - The opposite force that counterbalances the resultant force (in order to maintain equilibrium).
A state when objects are at rest or in uniform motion.
Example: Steady speed
NET FORCE is ZERO.
Forces
Forces are VECTORS.
GRAVITY - A very common example of a force that causes objects to accelerate at a rate of approximately 9.8 m/s2
FORCES cause an object to undergo acceleration.
Magnitude of a force is measured in newtons (N).
7.7 - Applications of the Dot Product & Cross Multiplication
Applications of the Cross Product
TORQUE
TORQUE is a VECTOR quantity measured in Newton-metres (N-m) or in joules (J).
Torque is caused by a FORCE defined as the CROSS PRODUCT in cases where force causes an object to TURN. (angular rather than linear displacement)
The axes are the cross products of each other since the only vector perpendicular to two axes is the third axis.
AREA OF A PARALLELOGRAM
Applications of Dot Product
WORK
CALCULATION
WORK is a SCALAR quantity measured in joules (J).
Work is done when a force acting on an object causes a DISPLACEMENT of an object from one position to another.
7.6 - The Cross Product of Two Vectors
Properties of the Cross Product
PROPERTY 3:
PROPERTY 2:
PROPERTY 1:
Formula for the Magnitude of the Cross Product
Formula for the Cross Product
Cross Product (Vector Product)
The CROSS PRODUCT is a VECTOR, and can only be defined in R3.
The CROSS PRODUCT of two vectors is the vector that is PERPENDICULAR to both of them.
The two non-collinear vectors and their cross product form a RIGHT-HANDED system.
7.5 - Scalar & Vector Projections
Direction Angles in R3
DIRECTION ANGLES are the angles that a position vector makes with each of the positive coordinate axes.
Vector Projections
Vector projections are a VECTOR (vector quantity)
Scalar Projections
CALCULATIONS
SPECIAL CASE
The general formula for finding the SCALAR PROJECTION of a vector on another vector:
There are 3 cases:
Case 3:
Case 2:
Case 1:
Scalar projections are a NUMBER (scalar quantity)
Projections
PROJECTIONS are formed from dropping a perpendicular from the head of one vector to another vector, or to the extension of another vector. (Similar to a SHADOW)
7.4 - The Dot Product of Algebraic Vectors
Property of the Dot Product in R3
R2 vs. R3
