Vectors - Problems - Algebra On 3 Vectors

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Vectors - Problems - Algebra on 3 vectors
Introduction to Algebra on 3 vectors
Solving problems using algebraic operations on vectors
Key concepts and formulas to be covered in this topic
Applications of algebra on 3 vectors in real-life situations

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Addition of two vectors algebraically
Components of vectors and their addition
Example: Addition of vectors A and B using algebraic methods
Equation: Resultant vector = Vector A + Vector B
Components: Resultant vector = ( (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} + (A_z + B_z) \hat{k} )

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Subtraction of two vectors algebraically
Components of vectors and their subtraction
Example: Subtraction of vector A from vector B using algebraic methods
Equation: Resultant vector = Vector B - Vector A
Components: Resultant vector = ( (B_x - A_x) \hat{i} + (B_y - A_y) \hat{j} + (B_z - A_z) \hat{k} )

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Scalar multiplication of a vector
Multiplying a vector by a scalar quantity
Example: Scalar multiplication of vector A by a scalar quantity ( \lambda )
Equation: Resultant vector = ( \lambda \cdot \text{Vector A} )
Components: Resultant vector = ( \lambda \cdot A_x \hat{i} + \lambda \cdot A_y \hat{j} + \lambda \cdot A_z \hat{k} )

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Dot product of two vectors
Formula for calculating dot product: ( \text{Vector A} \cdot \text{Vector B} = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z )
Example: Calculating dot product of vectors A and B
Geometrical interpretation of dot product
Applications of dot product in physics and engineering

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Cross product of two vectors
Formula for calculating cross product: ( \text{Vector A} \times \text{Vector B} = (A_y \cdot B_z - A_z \cdot B_y) \hat{i} - (A_x \cdot B_z - A_z \cdot B_x) \hat{j} + (A_x \cdot B_y - A_y \cdot B_x) \hat{k} )
Example: Calculating cross product of vectors A and B
Geometrical interpretation of cross product
Applications of cross product in physics and engineering

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Scalar triple product of three vectors
Formula for calculating scalar triple product: ( \text{Scalar Triple Product} = \text{Vector A} \cdot (\text{Vector B} \times \text{Vector C}) )
Example: Calculating scalar triple product of vectors A, B, and C
Geometrical interpretation of scalar triple product
Applications of scalar triple product in physics and engineering

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Vector triple product of three vectors
Formula for calculating vector triple product: ( \text{Vector Triple Product} = \text{Vector A} \times (\text{Vector B} \times \text{Vector C}) )
Example: Calculating vector triple product of vectors A, B, and C
Geometrical interpretation of vector triple product
Applications of vector triple product in physics and engineering

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Properties and identities of vectors
Commutative property of addition: ( \text{Vector A} + \text{Vector B} = \text{Vector B} + \text{Vector A} )
Associative property of addition: ( (\text{Vector A} + \text{Vector B}) + \text{Vector C} = \text{Vector A} + (\text{Vector B} + \text{Vector C}) )
Distributive property: ( \lambda \cdot (\text{Vector A} + \text{Vector B}) = \lambda \cdot \text{Vector A} + \lambda \cdot \text{Vector B} )

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Properties and identities of vectors (contd.)
Multiplicative property: ( \lambda \cdot (\mu \cdot \text{Vector A}) = (\lambda \cdot \mu) \cdot \text{Vector A} )
Zero vector property: ( \text{Vector A} + \text{Zero Vector} = \text{Vector A} )
Negative vector property: ( \text{Vector A} + (-\text{Vector A}) = \text{Zero Vector} )
Equality of vectors: Two vectors are equal if and only if their corresponding components are equal

Properties and identities of vectors (contd.)
Multiplicative property: ( \lambda \cdot (\mu \cdot \text{Vector A}) = (\lambda \cdot \mu) \cdot \text{Vector A} )
Zero vector property: ( \text{Vector A} + \text{Zero Vector} = \text{Vector A} )
Negative vector property: ( \text{Vector A} + (-\text{Vector A}) = \text{Zero Vector} )
Equality of vectors: Two vectors are equal if and only if their corresponding components are equal

Magnitude of a vector
Formula for calculating magnitude: (|\text{Vector A}| = \sqrt{A_x^2 + A_y^2 + A_z^2})
Example: Calculating the magnitude of vector A
Properties of magnitude: (|\text{Vector A}| \geq 0), (|\text{Vector A}| = 0) if and only if (\text{Vector A}) is the zero vector
Applications of magnitude in physics and engineering

Unit vector
Definition: A vector with magnitude 1 is called a unit vector
Formula for calculating unit vector: (\hat{u} = \frac{\text{Vector A}}{|\text{Vector A}|})
Example: Finding the unit vector in the direction of vector A
Properties of unit vectors
Applications of unit vectors in physics and engineering

Angle between two vectors
Formula for calculating the angle: ( \cos(\theta) = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector A}| \cdot |\text{Vector B}|})
Example: Calculating the angle between vectors A and B
Range of possible angles: 0° to 180°
Geometrical interpretation of the angle between vectors

Projection of a vector
Definition: The projection of vector A onto vector B is a vector parallel to B with magnitude ( |\text{Vector A}| \cdot \cos(\theta) ), where ( \theta ) is the angle between A and B
Formula for calculating the projection: ( \text{Projection of A onto B} = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector B}|^2} \cdot \text{Vector B} )
Example: Calculating the projection of vector A onto vector B
Geometrical interpretation of the projection
Applications of vector projections in physics and engineering

Component of a vector
Definition: The component of vector A along vector B is a vector parallel to B with magnitude ( |\text{Vector A}| \cdot \cos(\theta) ), where ( \theta ) is the angle between A and B
Formula for calculating the component: ( \text{Component of A along B} = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector B}|} \cdot \frac{\text{Vector B}}{|\text{Vector B}|})
Example: Calculating the component of vector A along vector B
Geometrical interpretation of the component
Applications of vector components in physics and engineering

Parallel vectors
Definition: Two vectors are parallel if they have the same direction or are in opposite directions
Testing for parallelism: If ( \text{Vector A} = \lambda \cdot \text{Vector B}) or ( \text{Vector A} = -\lambda \cdot \text{Vector B}), where ( \lambda ) is a scalar quantity, then A and B are parallel
Example: Determining whether vectors A and B are parallel
Applications of parallel vectors in physics and engineering

Collinear vectors
Definition: Three or more vectors are collinear if they lie on the same line or are parallel
Testing for collinearity: If there exist scalars ( \lambda_1, \lambda_2, \lambda_3 ) (not all zero) such that ( \text{Vector A} = \lambda_1 \cdot \text{Vector B} + \lambda_2 \cdot \text{Vector C} + \lambda_3 \cdot \text{Vector D} ), then A, B, C, and D are collinear
Example: Determining whether vectors A, B, and C are collinear
Applications of collinear vectors in physics and engineering

Perpendicular vectors
Definition: Two vectors are perpendicular if their dot product is zero
Testing for perpendicularity: If ( \text{Vector A} \cdot \text{Vector B} = 0 ), then A and B are perpendicular
Example: Determining whether vectors A and B are perpendicular
Geometrical interpretation of perpendicular vectors
Applications of perpendicular vectors in physics and engineering

Solving problems using algebra on 3 vectors
Strategies for solving vector problems algebraically
Identify known and unknown vectors
Break down vectors into components
Use appropriate algebraic operations to find the unknown vectors
Example: Solving a problem involving algebra on 3 vectors
Practice problems for further understanding and application of algebra on 3 vectors

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Geometric interpretation of dot product
Dot product as the product of magnitudes and the cosine of the angle between two vectors: ( \text{Vector A} \cdot \text{Vector B} = |\text{Vector A}| \cdot |\text{Vector B}| \cdot \cos(\theta) )
Properties of dot product: commutative, distributive, associative
Applications in finding the angle between vectors, determining if vectors are orthogonal, and solving problems involving work, displacement, and projections
Example: Find the dot product of vectors A = (3, 4) and B = (2, -1)

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Geometric interpretation of cross product
Cross product as the product of magnitudes, the sine of the angle between two vectors, and the unit normal vector to the plane they span: ( \text{Vector A} \times \text{Vector B} = |\text{Vector A}| \cdot |\text{Vector B}| \cdot \sin(\theta) \cdot \hat{n} )
Properties of cross product: non-commutative, distributive, anti-associative
Applications in finding the area of parallelograms, determining if vectors are coplanar, and solving problems involving torque, magnetic forces, and angular momentum
Example: Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6)

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Geometric interpretation of scalar triple product
Scalar triple product as the dot product of one vector with the cross product of the other two vectors: ( \text{Scalar Triple Product} = \text{Vector A} \cdot (\text{Vector B} \times \text{Vector C}) )
Applications in finding the volume of parallelepipeds, determining if vectors are coplanar, and solving problems involving forces, torques, and moments
Example: Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9)

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Commutative property of addition: ( \text{Vector A} + \text{Vector B} = \text{Vector B} + \text{Vector A} )
Associative property of addition: ( (\text{Vector A} + \text{Vector B}) + \text{Vector C} = \text{Vector A} + (\text{Vector B} + \text{Vector C}) )
Distributive property: ( \lambda \cdot (\text{Vector A} + \text{Vector B}) = \lambda \cdot \text{Vector A} + \lambda \cdot \text{Vector B} )
Multiplicative property: ( \lambda \cdot (\mu \cdot \text{Vector A}) = (\lambda \cdot \mu) \cdot \text{Vector A} )
Zero vector property: ( \text{Vector A} + \text{Zero Vector} = \text{Vector A} )

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Negative vector property: ( \text{Vector A} + (-\text{Vector A}) = \text{Zero Vector} )
Equality of vectors: Two vectors are equal if and only if their corresponding components are equal
Magnitude of a vector: (|\text{Vector A}| = \sqrt{A_x^2 + A_y^2 + A_z^2})
Example: Find the magnitude of vector A = (3, 4, 5)
Properties of magnitude: (|\text{Vector A}| \geq 0), (|\text{Vector A}| = 0) if and only if (\text{Vector A}) is the zero vector

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Unit vector: A vector with magnitude 1 is called a unit vector
Formula for calculating unit vector: (\hat{u} = \frac{\text{Vector A}}{|\text{Vector A}|})
Example: Find the unit vector in the direction of vector A = (3, 4, 5)
Properties of unit vectors
Applications of unit vectors in physics and engineering

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Angle between two vectors
Formula for calculating the angle: ( \cos(\theta) = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector A}| \cdot |\text{Vector B}|})
Example: Find the angle between vectors A = (3, 4) and B = (2, -1)
Range of possible angles: 0° to 180°
Geometric interpretation of the angle between vectors

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Projection of a vector
Definition: The projection of vector A onto vector B is a vector parallel to B with magnitude ( |\text{Vector A}| \cdot \cos(\theta) ), where ( \theta ) is the angle between A and B
Formula for calculating the projection: ( \text{Projection of A onto B} = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector B}|^2} \cdot \text{Vector B} )
Example: Calculate the projection of vector A = (3, 4) onto vector B = (2, -1)
Geometric interpretation of the projection
Applications of vector projections in physics and engineering

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Component of a vector
Definition: The component of vector A along vector B is a vector parallel to B with magnitude ( |\text{Vector A}| \cdot \cos(\theta) ), where ( \theta ) is the angle between A and B
Formula for calculating the component: ( \text{Component of A along B} = \frac{\text{Vector A} \cdot \text{Vector B}}{|\text{Vector B}|} \cdot \frac{\text{Vector B}}{|\text{Vector B}|})
Example: Calculate the component of vector A = (3, 4) along vector B = (2, -1)
Geometric interpretation of the component
Applications of vector components in physics and engineering

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Parallel vectors
Definition: Two vectors are parallel if they have the same direction or are in opposite directions
Testing for parallelism: If ( \text{Vector A} = \lambda \cdot \text{Vector B}) or ( \text{Vector A} = -\lambda \cdot \text{Vector B}), where ( \lambda ) is a scalar quantity, then A and B are parallel
Example: Determine whether vectors A = (3, 4) and B = (6, 8) are parallel
Applications of parallel vectors in physics and engineering