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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: $\cos(\theta) = \frac{A \cdot B}{|A||B|}$. Solve for $\theta$.
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: $comp_B A = \frac{A \cdot B}{|B|}$
Given two sides and the included angle of a triangle, how do you find the third side?
Use the Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos(C)$
Given two angles and a side of a triangle, how do you find the other sides?
Use the Law of Sines: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
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.
Explain scalar multiplication of a vector.
Changes the magnitude of the vector; reverses direction if the scalar is negative.
Explain vector addition.
Combining movements; add corresponding components. Geometrically, place tail of the second vector at the head of the first.
How does the dot product relate to the angle between vectors?
The dot product is related to the cosine of the angle between the vectors. A dot product of 0 implies the vectors are orthogonal.
What is the significance of a unit vector?
Represents direction without magnitude; useful for expressing any vector as a scaled version of a direction.
Explain the Triangle Inequality Theorem.
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
What does a negative dot product indicate?
The vectors point in generally opposite directions.
What does a positive dot product indicate?
The vectors point in generally similar directions.
How are vector components related to projections?
Vector components are the projections of the vector onto the x and y axes.
Explain the parallelogram rule for vector addition.
The resultant vector is the diagonal of the parallelogram formed by the two vectors being added.
What does it mean for a vector to be represented as a linear combination of i and j?
It means the vector is expressed as the sum of its x-component times the unit vector i and its y-component times the unit vector j.