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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
The equation of a vector in two dimensions can be written as a sum of its vertical and horizontal components, denoted βiβ and βjβ respectively. These are vectors that start from the origin and point right (βiβ) and up (βjβ), each with a length of β1. We call these base vectors.
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A vector in two dimensions can be uniquely described by giving its vertical and horizontal components, meaning the number of units in each direction required to get from the "tail" to the "head" of the arrow. These components can be represented using a column vector.
The column vector of a vector βvβ whose head is located βaβ horizontal and βbβ vertical units from its tail is
The top number represents the horizontal component and the bottom number represents the vertical component.
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A vector is a mathematical object that has both magnitude (size) and direction in space. A three-dimensional vector is typically described by its components along the βx, βy, and βzβ axes.
In 3 dimensions, a vector is expressed as
where βi, βj, and βkβ are called the base vectors:
To add two vectors, add their corresponding components:
To subtract two vectors, subtract their corresponding components:
Addition corresponds to placing vectors head-to-tail, and subtraction corresponds to the displacement between their tips.
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A position vector describes the position of a point relative to a fixed origin. For example, the position vector of a point βPβ describes the location of βPβ relative to an origin βO:
A displacement vector describes how to move from one point to another. For instance, the displacement vector from point βAβ to point βBβ is given by subtracting position vectors:
Thus, position vectors identify points relative to an origin, while displacement vectors represent movement or translation between points. Expressing βABβ as βOBβOAβ is sometimes referred to as giving the relative position of βBβ from βA.
The zero vector β0β is a special vector without size or a defined direction, represented by
The negative of a vector reverses the vector's direction while maintaining its size. If βv=βββxyzββ ββ, then the negative is
pointing in exactly the opposite direction.
Scaling a vector means multiplying it by a number βk, which changes its length without affecting its direction (unless βk<0, which reverses direction). If
then scaling by βkβ gives
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Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if βu=kvβ for some scalar βk. Parallel vectors have identical or exactly opposite directions.
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The magnitude (or length) of a vector βv=βββv1βv2βv3βββ βββ is calculated as
Geometrically, this represents the distance from the vector's starting point to its endpoint in 3D space. For example, the magnitude of the displacement vector βABβ equals the straight-line distance between points βAβ and βB.
When a vector βuβ has a magnitude of β1, we say βuβ is a unit vector.
We often express other vectors βvβ with magnitude ββ£vβ£ξ =1β as scalar multiples of the unit vector pointing in the same direction,
where βk=β£vβ£.
The unit vector in the same direction as a non-unit vector βvβ is written as
Rescaling to form a vector with a magnitude of one is referred to as normalizing a vector.
The scalar product, also called the dot product, takes two vectors and produces a scalar (a number). For two vectors βv=βββv1βv2βv3βββ βββ and βw=βββw1βw2βw3βββ ββ, the scalar product is calculated as:
This operation combines corresponding components of each vector, resulting in a single numerical value.
The scalar product also has a geometric interpretation involving the angle βΞΈβ between two vectors:
Equivalently, isolating βcosΞΈ:
The angle βΞΈβ is measured between the heads of βvβ and βw:
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If the scalar product of two vectors is negative, then
and thus βΞΈβ must be an obtuse angle: β90Β°<ΞΈβ€180Β°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
whenever βΞΈ>90Β°.
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Since βvβ w=β£vβ£β£wβ£cosΞΈ, if two vectors are perpendicular then βcos90Β°=0ββ their scalar product is β0.
For any vectors βu, βv, and βw, and scalar βk:
The vector product, sometimes called the cross product, of two vectors βvβ and βwβ is given by:
This vector is perpendicular to both original vectors. Using the right-hand rule, if your index finger points in the direction of βvβ and your middle finger points towards βw, your thumb points towards βvΓw. The vector product thus creates a new vector perpendicular to both original vectors.
The magnitude of the vector product βvΓwβ gives the area of the parallelogram formed by vectors βvβ and βw:
Intuitively, this happens because the magnitude combines both vectorsβ lengths and how "spread out" they are from each other, capturing exactly the amount of two-dimensional space they span.
The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula
Here, βΞΈβ is the angle between vectors βvβ and βw. This relationship holds because the area of the parallelogram formed by the two vectors depends on their lengths and the angle separating them.
Specifically, the area is largest when the vectors are perpendicular (βsin90β=1β) and zero when they are parallel (βsin0β=0β).
For two vectors βaβ and βb, the magnitude of the component of vector βaβ that acts in the direction of vector βbβ is given by
The magnitude of the component of vector βaβ that acts perpendicular to vector βb, in the plane formed by the two vectors, is given by
where βΞΈβ is the angle between the two vectors.
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For any vectors βu, βv, and βw, and scalar βk:
A vector line is defined by specifying one fixed point on the line and a direction vector. If the fixed point is given by position vector βa, and the direction vector by βb, then the line in vector form is:
Here, βΞ»βRβ is a parameter that varies over all real numbers, generating every point on the line. Changing βΞ»β moves the point βrβ along the direction of βb, creating the full infinite line.
A vector line in three-dimensional space can also be expressed in parametric form, showing explicitly the equations for the βx, βy, and βzβ coordinates separately:
Here, β(x0β,y0β,z0β)β is a point on the line, β(l,m,n)β are the components of the direction vector, and the parameter βΞ»β controls your position along the line. Changing βΞ»β moves the point continuously along the direction of the vector, producing the full infinite line.
In kinematics (the mathematical description of motion), 3D motion can be modeled by a vector line, often expressed with the parameter representing time, βt.
Specifically, the direction vector βbβ represents the velocity of an object, indicating both its direction and magnitude of movement. The magnitude of this vector, ββ£bβ£, is the object's speedβthe rate at which it moves, irrespective of direction.
The angle between two lines is simply the angle between their direction vectors.
For any two lines βr1β=a1β+Ξ»b1ββ and βr2β=a2β+ΞΌb2β, the angle βΞΈβ between βr1ββ and βr2ββ can be found via the formula
which is just the equation of the scalar product.