All Flashcards
How do you find the components of a vector given its magnitude and direction angle?
Calculate x-component: magnitude * cos(angle). Calculate y-component: magnitude * sin(angle).
How do you add two vectors given their components?
Add the corresponding x-components and y-components separately.
How do you find a unit vector in the same direction as a given vector?
Find the magnitude of the vector. Divide each component of the original vector by its magnitude.
How do you determine if two vectors are orthogonal?
Calculate their dot product. If the dot product is 0, the vectors are orthogonal.
How do you find the angle between two vectors?
Use the formula: . Solve for .
How do you find the resultant vector of two forces acting on an object?
Represent each force as a vector. Add the vectors component-wise to find the resultant vector.
How do you determine if three points can form a triangle?
Calculate the distances between each pair of points. Check if the Triangle Inequality Theorem holds.
How do you find the scalar projection of one vector onto another?
Use the formula:
Given two sides and the included angle of a triangle, how do you find the third side?
Use the Law of Cosines:
Given two angles and a side of a triangle, how do you find the other sides?
Use the Law of Sines:
What are the differences between scalar multiplication and dot product?
Scalar Multiplication: multiplies a vector by a scalar, resulting in a vector. | Dot Product: multiplies two vectors, resulting in a scalar.
What are the differences between vector addition and scalar addition?
Vector Addition: Combines vectors, considering both magnitude and direction. | Scalar Addition: Combines scalars, only considering magnitude.
What are the differences between Law of Sines and Law of Cosines?
Law of Sines: Relates sides to sines of opposite angles, useful with two angles and a side. | Law of Cosines: Relates sides and one angle, useful with three sides or two sides and included angle.
What are the differences between a vector and a scalar?
Vector: Has both magnitude and direction. | Scalar: Has only magnitude.
What are the differences between component-wise vector addition and the parallelogram rule?
Component-wise: Add corresponding components algebraically. | Parallelogram Rule: Geometric method involving drawing a parallelogram.
What are the differences between finding a unit vector and finding the magnitude of a vector?
Unit Vector: Normalizes a vector to length 1, preserving direction. | Magnitude: Calculates the length of the vector.
What are the differences between representing a vector geometrically and with components?
Geometrically: Represented as an arrow with length and direction. | Components: Represented as ordered pairs indicating x and y displacements.
What are the differences between the head and tail of a vector?
Head: The endpoint of the vector, indicating direction. | Tail: The starting point of the vector.
What are the differences between the x and y components of a vector?
X component: Projection of the vector onto the x-axis. | Y component: Projection of the vector onto the y-axis.
What are the differences between using sine and cosine to find vector components?
Sine: Used to find the y-component, opposite to the angle. | Cosine: Used to find the x-component, adjacent to the angle.
What is a vector?
An object with both magnitude and direction.
Define vector magnitude.
The length of the vector; a scalar quantity.
What is a scalar?
A quantity that has magnitude only, but no direction.
Define a unit vector.
A vector with a magnitude of 1.
What does orthogonal mean in the context of vectors?
Perpendicular; the dot product of orthogonal vectors is 0.
Define the 'tail' of a vector.
The starting point of the vector.
Define the 'head' of a vector.
The ending point of the vector.
What is the zero vector?
A vector with zero magnitude, denoted as <0, 0>.
Define vector components.
The x and y values that define a vector in a coordinate plane.
What is the dot product?
A scalar value resulting from the multiplication of two vectors, indicating their similarity in direction.