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Questions and Answers
If a vector A has a magnitude of 10 units and makes an angle of 30° with the positive x-axis, what is the y-component of the vector?
If a vector A has a magnitude of 10 units and makes an angle of 30° with the positive x-axis, what is the y-component of the vector?
- 8.66 units (correct)
- 5.77 units
- 10 units
- 5 units
A vector has x and y components of 4 units and 3 units, respectively. What is the magnitude of this vector?
A vector has x and y components of 4 units and 3 units, respectively. What is the magnitude of this vector?
- 5 units (correct)
- 7 units
- 12 units
- 1 unit
A vector has components $A_x = -5$ and $A_y = 5$. In which quadrant does this vector lie?
A vector has components $A_x = -5$ and $A_y = 5$. In which quadrant does this vector lie?
- Quadrant I
- Quadrant II (correct)
- Quadrant IV
- Quadrant III
Two vectors, A and B, have magnitudes of 5 and 8 units, respectively. They are oriented such that the angle between them is 60°. What is the magnitude of their resultant vector when using the parallelogram method?
Two vectors, A and B, have magnitudes of 5 and 8 units, respectively. They are oriented such that the angle between them is 60°. What is the magnitude of their resultant vector when using the parallelogram method?
Vector A has a magnitude of 7 units and points along the positive x-axis. Vector B has a magnitude of 9 units and points along the positive y-axis. Determine the direction of the resultant vector.
Vector A has a magnitude of 7 units and points along the positive x-axis. Vector B has a magnitude of 9 units and points along the positive y-axis. Determine the direction of the resultant vector.
A car travels 20 km east and then 30 km north. What is the magnitude of the car's total displacement?
A car travels 20 km east and then 30 km north. What is the magnitude of the car's total displacement?
Given two vectors: Vector A = 5i + 3j and Vector B = -2i + 6j, what is the resultant vector C = A + B?
Given two vectors: Vector A = 5i + 3j and Vector B = -2i + 6j, what is the resultant vector C = A + B?
Three vectors A, B, and C, lying in the xy-plane, are added together. The x-component of their sum is R_x and the y-component is R_y. What are $R_x$ and $R_y$?
Three vectors A, B, and C, lying in the xy-plane, are added together. The x-component of their sum is R_x and the y-component is R_y. What are $R_x$ and $R_y$?
A boat is rowed directly across a river 100 m wide. The boat's velocity relative to the water is 3.0 m/s. The river flows at a rate of 4.0 m/s. What is the magnitude of the resultant velocity of the boat?
A boat is rowed directly across a river 100 m wide. The boat's velocity relative to the water is 3.0 m/s. The river flows at a rate of 4.0 m/s. What is the magnitude of the resultant velocity of the boat?
Given vector $A = 4i - 2j$ and vector $B = -2i + 5j$, what is the direction of the resultant vector $A + B$ with respect to the positive x-axis?
Given vector $A = 4i - 2j$ and vector $B = -2i + 5j$, what is the direction of the resultant vector $A + B$ with respect to the positive x-axis?
Flashcards
Resolving a Vector
Resolving a Vector
Breaking down a vector into its x- and y-components using trigonometric ratios.
x-component of a Vector
x-component of a Vector
The component of a vector along the horizontal axis. It can be calculated using: 𝐴ₓ = 𝐴 cos θ (when θ is angle between vector and x-axis)
y-component of a Vector
y-component of a Vector
The component of a vector along the vertical axis. It can be calculated using: 𝐴y = 𝐴 sin θ (when θ is angle between vector and x-axis)
Component Method
Component Method
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Parallelogram Method
Parallelogram Method
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Law of Sines
Law of Sines
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Law of Cosines
Law of Cosines
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Resultant Magnitude
Resultant Magnitude
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Resultant Direction
Resultant Direction
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Study Notes
Resolving a Vector
- Vector components are determined using trigonometric ratios, known as resolving a vector into its components
- If the magnitude of vector A is known, along with the direction θ from the positive x-axis, the x and y components of vector A can be found using:
- cos θ = Ax/A → Ax = A cos θ
- sin θ = Ay/A → Ay = A sin θ
- The x and y components are given by Ax = A cos θ and Ay = A sin θ if the angle θ is between the vector and the x-axis
- If the components are known, the magnitude and direction of the vector can be found with:
- A² = Ax² + Ay²
- A = √(Ax² + Ay²)
- The direction is given by:
- tan θ = Ay/Ax
- θ = tan-1(Ay/Ax)
Adding Vectors Using Components
- Vectors A, B, and C are considered to find the resultant vector R
- Vector A is at a 30° angle with the positive x-axis, vector B is at a 40° angle with the negative x-axis, and C points toward the negative y-axis
- Draw the vectors with their tails at the origin and resolve each vector into its x and y components
- Add all the x-components together and all the y-components together
- Ax = +A cos 30°
- Bx = -B cos 40°
- Cx = +C cos 90° = 0
- Rx = Ax + (-Bx) + Cx
- Ay = +A sin 30°
- By = +B sin 40°
- Cy = -C sin 0° = -C
- Ry = Ay + By + (-Cy)
- The magnitude of the resultant vector R is computed from Rx and Ry
- R² = Rx² + Ry²
- R = √(Rx² + Ry²)
- The direction of the resultant vector from Rx and Ry is found with:
- θ = tan-1 (Ry/Rx)
Vector Addition (Example)
- Find the resultant of three vectors where A and B are perpendicular, with magnitudes of A, B, and C being 10, 20, and 15 units, respectively
- Find the resultant of two vectors where vector A is at an angle of -25° with the positive x-axis and vector B is at an angle of 40° with the negative x-axis
- Vector A is at 50 m and vector B is at 30 m
- The Resultant vector R = 22.4 m and θ = 4.6°
Adding Vectors (Parallelogram Method)
- Two given vectors A and B at an angle θ to each other
- Construct lines parallel/equal to A and B to obtain a parallelogram
- The resultant R is the diagonal
- The co-interior angle sum equals 180°, so Φ = 180° – θ
- Cosine Rule: used to determine the magnitudes of A, B, and Φ
- Sine Rule: used to determine σ
Parallelogram Method (Example)
- A car travels east for 10.0 km, then travels 5.00 km in a 60° north of east direction
- Find displacement using the parallelogram method
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Description
Learn how to resolve vectors into components using trigonometric ratios. Understand how to find x and y components, and calculate magnitude and direction. Explore vector addition using components with examples involving angles and axes.
