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Questions and Answers
Which of the following is classified as a base quantity?
Which of the following is classified as a base quantity?
Which of the following is an example of a derived quantity?
Which of the following is an example of a derived quantity?
Which quantities can be classified as derived and vector?
Which quantities can be classified as derived and vector?
What distinguishes a scalar quantity from a vector quantity?
What distinguishes a scalar quantity from a vector quantity?
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Which of the following pairs correctly categorizes pressure?
Which of the following pairs correctly categorizes pressure?
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What is the correct classification of the quantities speed and work?
What is the correct classification of the quantities speed and work?
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Which physical quantity is classified as a base quantity?
Which physical quantity is classified as a base quantity?
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Which statement best explains derived quantities?
Which statement best explains derived quantities?
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What is the relationship defined by the equation $sin^2 q + cos^2 q = 1$?
What is the relationship defined by the equation $sin^2 q + cos^2 q = 1$?
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If $tan q = \frac{perpendicular}{base}$, which of the following is $cot q$?
If $tan q = \frac{perpendicular}{base}$, which of the following is $cot q$?
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Which of these represents the trigonometric ratio for $sec q$?
Which of these represents the trigonometric ratio for $sec q$?
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What is the value of $cosec q$ if $sin q = \frac{3}{5}$?
What is the value of $cosec q$ if $sin q = \frac{3}{5}$?
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Which equation correctly relates $cot^2 q$ to $cosec^2 q$?
Which equation correctly relates $cot^2 q$ to $cosec^2 q$?
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What is the angle at which the magnitude of the resultant of two vectors is maximized?
What is the angle at which the magnitude of the resultant of two vectors is maximized?
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In which case is the magnitude of the resultant minimum?
In which case is the magnitude of the resultant minimum?
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When adding two vectors with the same magnitude, what is the resultant when the angle between them is 120 degrees?
When adding two vectors with the same magnitude, what is the resultant when the angle between them is 120 degrees?
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What will happen to the resultant magnitude if vector Q is reversed before summing with P?
What will happen to the resultant magnitude if vector Q is reversed before summing with P?
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If two vectors of equal magnitude are arranged at 90 degrees, what is the magnitude of their resultant?
If two vectors of equal magnitude are arranged at 90 degrees, what is the magnitude of their resultant?
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What is the relationship between the angles of Case II and the maximum magnitude of the resultant?
What is the relationship between the angles of Case II and the maximum magnitude of the resultant?
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If the resultant of two vectors is 15 units when they are added directly, how does this change when one vector is reversed?
If the resultant of two vectors is 15 units when they are added directly, how does this change when one vector is reversed?
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Which of the following cases provides the largest resultant vector?
Which of the following cases provides the largest resultant vector?
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What operation is performed when subtracting vector B from vector A?
What operation is performed when subtracting vector B from vector A?
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Which of the following properties of vector addition describes that the order of addition does not change the result?
Which of the following properties of vector addition describes that the order of addition does not change the result?
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If two vectors A and B form an angle of 0°, what is the resulting magnitude of their resultant R?
If two vectors A and B form an angle of 0°, what is the resulting magnitude of their resultant R?
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What does the expression |A ± B| represent?
What does the expression |A ± B| represent?
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When the angle between vectors A and B is 120°, what can be said about their magnitudes and resultant?
When the angle between vectors A and B is 120°, what can be said about their magnitudes and resultant?
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Which formula represents the magnitude of the resultant vector when two vectors A and B are at angle q?
Which formula represents the magnitude of the resultant vector when two vectors A and B are at angle q?
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What does the angle between two perpendicular vectors A and B equate to?
What does the angle between two perpendicular vectors A and B equate to?
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What do you get when you calculate the component of vector A in the direction of vector B?
What do you get when you calculate the component of vector A in the direction of vector B?
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What is A^, the component of A perpendicular to B?
What is A^, the component of A perpendicular to B?
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Which of the following statements about vector addition is true?
Which of the following statements about vector addition is true?
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In the dot product of two vectors A and B, what is represented by A·B?
In the dot product of two vectors A and B, what is represented by A·B?
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What term describes the operation where vector B is added in the negative direction?
What term describes the operation where vector B is added in the negative direction?
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What is the condition for the dot product of two Cartesian unit vectors to equal zero?
What is the condition for the dot product of two Cartesian unit vectors to equal zero?
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What is the result when the angle between two vectors A and B is π radians?
What is the result when the angle between two vectors A and B is π radians?
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Which of the following products is equal to 1 when calculating the dot product of Cartesian unit vectors?
Which of the following products is equal to 1 when calculating the dot product of Cartesian unit vectors?
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What does the equation A × A = A² imply about the properties of vector A?
What does the equation A × A = A² imply about the properties of vector A?
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If A is represented as A = Ax ˆi + Ay ˆj + Az ˆk, what are the components Ax, Ay, and Az representing?
If A is represented as A = Ax ˆi + Ay ˆj + Az ˆk, what are the components Ax, Ay, and Az representing?
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When calculating the angle θ between vectors A and B, which formula would be used?
When calculating the angle θ between vectors A and B, which formula would be used?
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What does the notation A |= A cos θ represent in vector analysis?
What does the notation A |= A cos θ represent in vector analysis?
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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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Description
Test your knowledge on vector operations including dot products and vector components. This quiz covers the fundamentals necessary to understand vectors in physics and mathematics. Challenge yourself with problems on finding angles, magnitudes, and resultant vectors.
