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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.

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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.

Formula for vector components given two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ .

$<x_2 - x_1, y_2 - y_1>$

Magnitude of vector $[object Object]$ ?

$\sqrt{a^2 + b^2}$

Dot product of vectors $A = <A_x, A_y>$ and $B = <B_x, B_y>$ ?

$A \cdot B = (A_x * B_x) + (A_y * B_y)$

How to find a unit vector?

$\frac{\vec{v}}{|\vec{v}|}$

Law of Sines formula.

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

Law of Cosines formula to find side 'a'.

$a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$

How to represent vector $[object Object]$ using standard unit vectors?

$ai + bj$

If given magnitude |v| and angle θ, what is the x-component of the vector?

$|v| \cos(\theta)$

If given magnitude |v| and angle θ, what is the y-component of the vector?

$|v| \sin(\theta)$

Formula for the magnitude of the resultant vector when adding two vectors.

$|\vec{p} + \vec{q}|^2 = |\vec{p}|^2 + |\vec{q}|^2 + 2|\vec{p}||\vec{q}|cos(\theta)$

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.