Physics: Rectangular Components of a Vector

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Questions and Answers

What is the primary characteristic of the scalar product (dot product) of two vectors?

  • It results in a vector quantity.
  • It results in a scalar quantity. (correct)
  • It depends on the angle between the vectors.
  • It always equals zero.

The x-component and y-component of a vector are always parallel to each other.

False (B)

What is used to find the direction angle θ of a vector?

tan θ = Ay / Ax

The __ product of two vectors results in a vector quantity.

<p>cross</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Dot Product = Results in a scalar quantity Cross Product = Results in a vector quantity Magnitude of Vector = Length of the vector Direction Angle = Angle made with the horizontal</p> Signup and view all the answers

How do you compute the magnitude 'A' of a vector from its rectangular components Ax and Ay?

<p>A = √(Ax^2 + Ay^2) (B)</p> Signup and view all the answers

The components of a vector can only be calculated using angles larger than 90 degrees.

<p>False (B)</p> Signup and view all the answers

What is the relationship between the unit vectors i and j?

<p>i represents the x-axis direction, and j represents the y-axis direction.</p> Signup and view all the answers

What is the definition of the vector product of two quantities?

<p>It produces a vector quantity. (A)</p> Signup and view all the answers

The scalar product of orthogonal vectors is equal to one.

<p>False (B)</p> Signup and view all the answers

What is the scalar product of parallel vectors A and B when the angle between them is 0°?

<p>A·B = |A| |B|</p> Signup and view all the answers

The scalar product of a vector with itself yields the square of its _____ .

<p>magnitude</p> Signup and view all the answers

Match the following properties of the scalar product with their definitions:

<p>Commutative Property = A·B = B·A Scalar Product of Orthogonal Vectors = A·B = 0 Scalar Product of a Unit Vector = i·i = 1 Scalar Product with Antiparallel Vectors = A·B = -|A| |B|</p> Signup and view all the answers

What is notable about the scalar product of a vector with the null vector?

<p>It is equal to zero. (B)</p> Signup and view all the answers

The dot product of a vector with itself is greater than zero.

<p>True (A)</p> Signup and view all the answers

What is the result of the dot product of two antiparallel vectors?

<p>Negative value</p> Signup and view all the answers

What is the formula for the cross product of two vectors A and B?

<p>A × B = |A| |B| sin(θ) n̂ (C)</p> Signup and view all the answers

The vector product of two parallel vectors results in a non-zero vector.

<p>False (B)</p> Signup and view all the answers

What rule is used to determine the direction of the cross product?

<p>Right-hand rule</p> Signup and view all the answers

Torque is an example of a vector product, expressed as τ = r × F, where r is the position vector and F is the _______.

<p>force</p> Signup and view all the answers

Match each term with its correct definition:

<p>Torque = The vector product of force and position vector Angular Momentum = The cross product of linear momentum and position vector Perpendicular Vectors = Vectors that create maximum vector product Anti-commutative Property = A × B = -B × A</p> Signup and view all the answers

Which of the following statements correctly describes the anti-commutative property of the vector product?

<p>A × B = -B × A (A)</p> Signup and view all the answers

The vector product of two perpendicular vectors is zero.

<p>False (B)</p> Signup and view all the answers

What happens to the cross product if one of the vectors is the same as the other?

<p>It equals zero.</p> Signup and view all the answers

Flashcards

Cross Product Definition

The cross product of two vectors A and B (A × B) is a vector perpendicular to both A and B, with a magnitude equal to the product of the magnitudes of A and B, and the sine of the angle between them.

Cross Product Magnitude

The magnitude of the cross product (A × B) is given by |A| |B| sin(θ), where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

Cross Product Direction

The direction of the cross product is determined by the right-hand rule; rotate your right hand fingers from vector A to vector B. Your thumb points in the direction of the cross product.

Non-commutative Cross Product

The cross product is anti-commutative, meaning A × B = -B × A.

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Parallel/Antiparallel Vectors

The cross product of two parallel or antiparallel vectors is zero (A × B = 0).

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Cross Product of Itself

The cross product of a vector with itself is zero (A × A = 0).

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Torque

Torque (τ) is the cross product of the position vector (r) and the force vector (F), τ = r × F.

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Angular Momentum

Angular momentum (L) is the cross product of linear momentum (p) and the position vector (r), L = r × p.

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Dot Product Definition

The dot product of two vectors A and B, denoted by A·B, is the product of the magnitudes of the two vectors and the cosine of the angle between them.

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Dot Product Commutative Property

The dot product of two vectors is commutative, meaning A·B = B·A.

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Orthogonal Vectors Dot Product

The dot product of two orthogonal (perpendicular) vectors is zero.

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Parallel Vectors Dot Product

The dot product of two parallel vectors is the product of their magnitudes.

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Vector Dot Product with Itself

The dot product of a vector with itself is equal to the square of its magnitude.

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Unit Vector Dot Product with Itself

The dot product of a unit vector with itself is 1.

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Dot Product with Null Vector

The dot product of a vector with the null vector is zero.

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Antiparallel Vectors Dot Product

The dot product of two antiparallel vectors is the negative of the product of their magnitudes.

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Vector Components

A vector can be broken down into its individual parts along perpendicular axes, called components. These components represent the vector's effect in each direction.

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Rectangular Components

In 2D, a vector's components are usually along the x-axis (horizontal) and y-axis (vertical), forming a right angle. They are called rectangular components.

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Finding x-Component

The x-component of a vector is found by projecting the vector onto the x-axis. This projection is the length of the side adjacent to the angle between the vector and x-axis.

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Finding y-Component

The y-component of a vector is found by projecting the vector onto the y-axis. This projection is the length of the side opposite the angle between the vector and x-axis.

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Vector Magnitude from Components

The magnitude (length) of a vector can be found using the Pythagorean theorem: square the x-component, square the y-component, add them together, and take the square root.

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Vector Direction from Components

The direction of a vector (angle it makes with the x-axis) is found using the arctangent (tan⁻¹) function: divide the y-component by the x-component and take the inverse tangent.

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Dot Product

The dot product of two vectors is a scalar quantity obtained by multiplying the magnitudes of the vectors and the cosine of the angle between them.

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Dot Product Formula

The dot product of vectors A and B is represented as A · B = |A| |B| cos θ, where θ is the angle between them.

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Study Notes

Rectangular Components of a Vector

  • Components are parts of a vector in different directions.
  • In 2D, vectors have x and y components.
  • The x-component is the horizontal component.
  • The y-component is the vertical component.
  • X and y components are perpendicular to each other.
  • Rectangular components are perpendicular components.

Rectangular Components

  • A vector 'A' makes an angle 'θ' with the horizontal.
  • The x-component (Ax) is the projection of vector A along the x-axis.
  • The y-component (Ay) is the projection of vector A along the y-axis.
  • Vector A can be written as A = Ax + Ay or A = Axi + Ayj
  • i and j are unit vectors in the x and y directions, respectively.

Finding Rectangular Components

  • Trigonometric ratios are used to find magnitudes of rectangular components (Ax and Ay).
  • Consider a right-angled triangle (OPO).
  • sin θ = perpendicular/hypotenuse
  • Calculations for Ax and Ay depend on the angle θ in the triangle.

Scalar Product (Dot Product)

  • The product of two vector quantities can be a scalar or vector.
  • A scalar product (dot product) results in a scalar quantity.
  • The scalar product of two vectors A and B is written as A.B.
  • A.B = |A| |B| cos θ (where θ is the angle between A and B).
  • The dot product is commutative (A.B = B.A).

Properties of Scalar Product

  • The scalar product of two perpendicular vectors is 0.
  • The scalar product of a vector with itself is equal to the square of its magnitude.
  • The scalar product of two parallel vectors is equal to the product of their magnitudes.
  • The scalar product of a vector with the null vector is 0.

Vector Product (Cross Product)

  • The product of two vector quantities can be a scalar or vector quantity.
  • A vector product results in a vector quantity.
  • The vector product of two vectors A and B is written as A×B.
  • A×B = |A| |B| sin θ n (where θ is the angle between A and B, and n is a unit vector perpendicular to both A and B).
  • Direction of the cross product is found using the right-hand rule.

Properties of the Vector Product

  • The vector product is anti-commutative: A×B = -B×A.
  • The vector product of two parallel vectors is zero.
  • The vector product of a vector with itself is zero.

Vector Product (in terms of rectangular components)

  • The vector product of two vectors 'A' and 'B' can be represented using their rectangular components.
  • The vector product of two vectors in rectangular components is determined using a determinant.

Significance of Vector Product

  • Calculating the area of a parallelogram or triangle using vectors involving their magnitudes and the included angle.

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