Vector Operations Quiz

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Questions and Answers

Which of the following is classified as a base quantity?

  • Force
  • Mass (correct)
  • Velocity
  • Pressure

Which of the following is an example of a derived quantity?

  • Distance
  • Time
  • Speed (correct)
  • Mass

Which quantities can be classified as derived and vector?

  • Mass and displacement
  • Time and work
  • Force and acceleration (correct)
  • Speed and pressure

What distinguishes a scalar quantity from a vector quantity?

<p>Scalars have magnitude only, vectors have both magnitude and direction. (D)</p> Signup and view all the answers

Which of the following pairs correctly categorizes pressure?

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

What is the correct classification of the quantities speed and work?

<p>Both are derived quantities (C)</p> Signup and view all the answers

Which physical quantity is classified as a base quantity?

<p>Time (C)</p> Signup and view all the answers

Which statement best explains derived quantities?

<p>Derived quantities are dependent on base quantities and can be expressed from them. (D)</p> Signup and view all the answers

What is the relationship defined by the equation $sin^2 q + cos^2 q = 1$?

<p>It is a trigonometric identity. (C)</p> Signup and view all the answers

If $tan q = \frac{perpendicular}{base}$, which of the following is $cot q$?

<p>The base divided by the perpendicular. (B)</p> Signup and view all the answers

Which of these represents the trigonometric ratio for $sec q$?

<p>Hypotenuse divided by the perpendicular. (B)</p> Signup and view all the answers

What is the value of $cosec q$ if $sin q = \frac{3}{5}$?

<p>$\frac{5}{3}$ (B)</p> Signup and view all the answers

Which equation correctly relates $cot^2 q$ to $cosec^2 q$?

<p>$1 + cot^2 q = cosec^2 q$ (B)</p> Signup and view all the answers

What is the angle at which the magnitude of the resultant of two vectors is maximized?

<p>30 degrees (A)</p> Signup and view all the answers

In which case is the magnitude of the resultant minimum?

<p>Case I (D)</p> Signup and view all the answers

When adding two vectors with the same magnitude, what is the resultant when the angle between them is 120 degrees?

<p>2 units (B)</p> Signup and view all the answers

What will happen to the resultant magnitude if vector Q is reversed before summing with P?

<p>It increases (D)</p> Signup and view all the answers

If two vectors of equal magnitude are arranged at 90 degrees, what is the magnitude of their resultant?

<p>$\sqrt{2}A$ (C)</p> Signup and view all the answers

What is the relationship between the angles of Case II and the maximum magnitude of the resultant?

<p>They are supplementary (A)</p> Signup and view all the answers

If the resultant of two vectors is 15 units when they are added directly, how does this change when one vector is reversed?

<p>Decreases to 113 units (D)</p> Signup and view all the answers

Which of the following cases provides the largest resultant vector?

<p>Case II (C)</p> Signup and view all the answers

What operation is performed when subtracting vector B from vector A?

<p>Add the negative vector –B to vector A (C)</p> Signup and view all the answers

Which of the following properties of vector addition describes that the order of addition does not change the result?

<p>Commutative Property (C)</p> Signup and view all the answers

If two vectors A and B form an angle of 0°, what is the resulting magnitude of their resultant R?

<p>|R| = |A| + |B| (B)</p> Signup and view all the answers

What does the expression |A ± B| represent?

<p>The magnitude of the resulting vector from adding or subtracting A and B (D)</p> Signup and view all the answers

When the angle between vectors A and B is 120°, what can be said about their magnitudes and resultant?

<p>|R| = |A| (D)</p> Signup and view all the answers

Which formula represents the magnitude of the resultant vector when two vectors A and B are at angle q?

<p>|R| = √(|A|^2 + |B|^2 ± 2|A||B| cos(q)) (A)</p> Signup and view all the answers

What does the angle between two perpendicular vectors A and B equate to?

<p>90° (A)</p> Signup and view all the answers

What do you get when you calculate the component of vector A in the direction of vector B?

<p>A| = A cos θ (D)</p> Signup and view all the answers

What is A^, the component of A perpendicular to B?

<p>A - A| (D)</p> Signup and view all the answers

Which of the following statements about vector addition is true?

<p>The sum of any two vectors can be computed using their components. (D)</p> Signup and view all the answers

In the dot product of two vectors A and B, what is represented by A·B?

<p>Ax Bx + Ay By + Az Bz (D)</p> Signup and view all the answers

What term describes the operation where vector B is added in the negative direction?

<p>Subtraction (D)</p> Signup and view all the answers

What is the condition for the dot product of two Cartesian unit vectors to equal zero?

<p>When the vectors are perpendicular (A)</p> Signup and view all the answers

What is the result when the angle between two vectors A and B is π radians?

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

Which of the following products is equal to 1 when calculating the dot product of Cartesian unit vectors?

<p>ˆk × ˆk (A), ˆj × ˆj (B)</p> Signup and view all the answers

What does the equation A × A = A² imply about the properties of vector A?

<p>A's magnitude squared is equal to its self-dot product (B)</p> Signup and view all the answers

If A is represented as A = Ax ˆi + Ay ˆj + Az ˆk, what are the components Ax, Ay, and Az representing?

<p>The x, y, and z components of vector A respectively (A)</p> Signup and view all the answers

When calculating the angle θ between vectors A and B, which formula would be used?

<p>θ = cos^-1(A·B/A|B|) (D)</p> Signup and view all the answers

What does the notation A |= A cos θ represent in vector analysis?

<p>The scalar projection of A onto B (A)</p> Signup and view all the answers

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Study Notes

Vector Operations

  • The angle between vectors A and B is given by:
    • q = arccos((A • B) / (||A|| ||B||))
  • The component of vector A in the direction of vector B is given by:
    • A|| = ( A • B / ||B||2) B
  • The component of vector A perpendicular to vector B is given by:
    • A^ = A - A||
  • The dot product of Cartesian unit vectors:
    • i • i = j • j = k • k = 1
    • i • j = j • k = k • i = 0
  • If A = Axi + Ayj + Azk and B = Bxi + Byj + Bzk, their dot product is given by:
    • A • B = AxBx + AyBy + AzBz

Resultant Vector Magnitude

  • The magnitude of the resultant vector R of two vectors A and B is given by:
    • ||R|| = √(||A||2 + ||B||2 + 2||A|| ||B|| cos q)
    • where q is the angle between A and B.

Special Cases of Vector Addition

  • When the angle between A and B is 0° (parallel):
    • ||R|| = ||A|| + ||B||
    • ||R|| is maximum
  • When the angle between A and B is 180° (anti-parallel):
    • ||R|| = ||A|| - ||B||
    • ||R|| is minimum
  • When the angle between A and B is 90° (perpendicular):
    • ||R|| = √(||A||2 + ||B||2)
  • For equal magnitude vectors (||A|| = ||B|| = a) with an angle of 120°:
    • ||R|| = a

Fundamental and Derived Quantities

  • Fundamental quantities:
    • Independent physical quantities that form the basis for all other quantities.
    • Examples: Mass, time
  • Derived quantities:
    • Quantities expressed in terms of fundamental quantities.
    • Examples: Speed, volume

Trigonometric Ratios

  • The following ratios of a right-angled triangle are called trigonometric ratios:
    • sin q = (opposite side) / (hypotenuse)
    • cos q = (adjacent side) / (hypotenuse)
    • tan q = (opposite side) / (adjacent side)
    • cot q = (adjacent side) / (opposite side)
    • sec q = (hypotenuse) / (adjacent side)
    • cosec q = (hypotenuse) / (opposite side)

Trigonometric Identities

  • sin2q + cos2q = 1
  • 1 + tan2q = sec2q
  • 1 + cot2q = cosec2q

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