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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 i and j respectively. These are vectors that start from the origin and point right (i) and up (j), each with a length of 1. 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 v whose head is located a horizontal and b 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 x, y, and z axes.
In 3 dimensions, a vector is expressed as
where i, j, and k 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 P describes the location of P relative to an origin O:
A displacement vector describes how to move from one point to another. For instance, the displacement vector from point A to point B 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 AB as OBβOA is sometimes referred to as giving the relative position of B from A.
The zero vector 0 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 v=βββxyzββ ββ, then the negative is
pointing in exactly the opposite direction.
Scaling a vector means multiplying it by a number k, which changes its length without affecting its direction (unless k<0, which reverses direction). If
then scaling by k gives
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Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if u=kv for some scalar k. Parallel vectors have identical or exactly opposite directions.
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The magnitude (or length) of a vector v=βββv1βv2βv3βββ ββ 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 AB equals the straight-line distance between points A and B.
When a vector u has a magnitude of 1, we say u is a unit vector.
We often express other vectors v with magnitude β£vβ£ξ =1 as scalar multiples of the unit vector pointing in the same direction,
where k=β£vβ£.
The unit vector in the same direction as a non-unit vector v 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 v=βββv1βv2βv3βββ ββ and w=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 ΞΈ between two vectors:
Equivalently, isolating cosΞΈ:
The angle ΞΈ is measured between the heads of v and w:
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If the scalar product of two vectors is negative, then
and thus ΞΈ must be an obtuse angle: 90Β°<ΞΈβ€180Β°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
whenever ΞΈ>90Β°.
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Since vβ w=β£vβ£β£wβ£cosΞΈ, if two vectors are perpendicular then cos90Β°=0β their scalar product is 0.
For any vectors u, v, and w, and scalar k:
The vector product, sometimes called the cross product, of two vectors v and w 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 v and your middle finger points towards w, your thumb points towards vΓw. The vector product thus creates a new vector perpendicular to both original vectors.
The magnitude of the vector product vΓw gives the area of the parallelogram formed by vectors v and w:
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, ΞΈ is the angle between vectors v and w. 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 (sin90β=1) and zero when they are parallel (sin0β=0).
For two vectors a and b, the magnitude of the component of vector a that acts in the direction of vector b is given by
The magnitude of the component of vector a that acts perpendicular to vector b, in the plane formed by the two vectors, is given by
where ΞΈ is the angle between the two vectors.
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For any vectors u, v, and w, and scalar k:
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 a, and the direction vector by b, then the line in vector form is:
Here, Ξ»βR is a parameter that varies over all real numbers, generating every point on the line. Changing Ξ» moves the point r along the direction of b, 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 x, y, and z coordinates separately:
Here, (x0β,y0β,z0β) is a point on the line, (l,m,n) are the components of the direction vector, and the parameter Ξ» controls your position along the line. Changing Ξ» 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, t.
Specifically, the direction vector b represents the velocity of an object, indicating both its direction and magnitude of movement. The magnitude of this vector, β£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 r1β=a1β+Ξ»b1β and r2β=a2β+ΞΌb2β, the angle ΞΈ between r1β and r2β can be found via the formula
which is just the equation of the scalar product.