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  2. AP Pre Calculus
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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 P1(x1,y1)P_1(x_1, y_1)P1​(x1​,y1​) and P2(x2,y2)P_2(x_2, y_2)P2​(x2​,y2​).

<x2−x1,y2−y1><x_2 - x_1, y_2 - y_1><x2​−x1​,y2​−y1​>

Magnitude of vector [objectObject][object Object][objectObject]?

a2+b2\sqrt{a^2 + b^2}a2+b2​

Dot product of vectors A=<Ax,Ay>A = <A_x, A_y>A=<Ax​,Ay​> and B=<Bx,By>B = <B_x, B_y>B=<Bx​,By​>?

A⋅B=(Ax∗Bx)+(Ay∗By)A \cdot B = (A_x * B_x) + (A_y * B_y)A⋅B=(Ax​∗Bx​)+(Ay​∗By​)

How to find a unit vector?

v⃗∣v⃗∣\frac{\vec{v}}{|\vec{v}|}∣v∣v​

Law of Sines formula.

asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}sin(A)a​=sin(B)b​=sin(C)c​

Law of Cosines formula to find side 'a'.

a2=b2+c2−2bc⋅cos⁡(A)a^2 = b^2 + c^2 - 2bc \cdot \cos(A)a2=b2+c2−2bc⋅cos(A)

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

ai+bjai + bjai+bj

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

∣v∣cos⁡(θ)|v| \cos(\theta)∣v∣cos(θ)

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

∣v∣sin⁡(θ)|v| \sin(\theta)∣v∣sin(θ)

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

∣p⃗+q⃗∣2=∣p⃗∣2+∣q⃗∣2+2∣p⃗∣∣q⃗∣cos(θ)|\vec{p} + \vec{q}|^2 = |\vec{p}|^2 + |\vec{q}|^2 + 2|\vec{p}||\vec{q}|cos(\theta)∣p​+q​∣2=∣p​∣2+∣q​∣2+2∣p​∣∣q​∣cos(θ)

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.