Vectors

Important Notes on Vectors

- the dot product of orthogonal vectors = 0 - the cross product of parallel/collinear vectors (which is same as cross product of a vector with itself) = 0 - the corresponding components of parallel/collinear vectors have the same ratio. - the dot product of two vectors is a scalar and lies in the plane of the two vectors. - the cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.

Types of Vectors

1) Zero vector - has 0 magnitude and denoted as 0->=(0,0,0) 2) Unit vector - has a magnitude of 1. Notation of unit vectors has a ^ on top of the vector name (A, B, C, etc.). To convert a "normal vector" into a unit vector, just divide the components of the normal vector by the vector's magnitude. 3) Co-initial vector - two or more vectors with same initial point 4) Parallel (aka collinear) vectors - two or more vectors with same or different magnitudes lying on the same line or on parallel lines going in the same direction (angle is = 0). They may be identified easily as components of one vector are a multiple of the other vector (e.g. (1,3) and (3,9) are parallel vectors with a multiple of 3. The cross product of parallel (collinear) vectors = 0 4a) Anti-parallel vectors - two or more vectors with same or different magnitudes lying on the same line or on parallel lines going in opposite directions (angle is = 180). They may be identified easily as components of one vector are a multiple of the other vector (e.g. (1,3) and (3,9) are parallel vectors with a multiple of 3. The cross product of anti-parallel vectors = 0 5) Equal vectors - two or more vectors with the same magnitude and direction (not required to have the same origin and terminus) 6) Negative vectors - vectors with the same magnitude but opposite direction 7) Orthogonal vectors - vectors whose angle between them is 90 degrees. The dot product of orthogonal vectors = 0. For vectors A->=(-2,4) and B->=(-16,-8):

Method to Add Vectors on a Graph

1) decide on scale factor of the vector magnitude 2) using a ruler, draw the 1st vector in the direction indicated. The length of the vector should be the scaled length of the vector. 3) use a protractor to plot the angle between vectors, then draw the second vector at that angle. Be sure to put the tail of one vector against the tip of the other vector. 4) draw the resultant from the unconnected tail of one vector to the unconnected tip of the other vector. 5) use a ruler to measure the length of the resultant from step 4 6) multiply the resultants length by the scale factor chosen in step 1. 7) use a protractor to measure the angle that the resultant makes with the positive x-axis.

Dot Product of Vectors

A measure of how closely two vectors align in terms of the directions they point. Dot product - Option 1) Used when vectors AND angle are given. Formula: A->.B-> = |A|*|B|*cosɵ * Option 2) Used when angle is NOT given. Formula: A->. B-> = AxBx + AyBy + AzBz

Vector

A quantity that has magnitude and direction. For example, 20m/s north is a vector. 20m/s is only velocity, not a vector, because it has magnitude but not direction. A vector of 5 at 45 degrees means the magnitude of the vector is 5 at a 45 degree angle.

The multiplication of a vector with a scalar λ does what?

A scalar changes the magnitude of the vector as a multiple of the scalar, but it does not change the direction of the vector. λA→=λa1^i+λb1^j+λc1^kλ λ|A| = λ|A|

Resultant

A vector that represents the sum of two or more vectors

Formula for the length of the x component (aka as "i") of vector A

A(subx) = |A|*cosɵ

Formula for the length of the y component (aka as "j") of vector A

A(suby) = |A|*sinɵ

Compare and contract dot product vs. cross product.

Dot product: - A.B = B.A - dot products are scalar values, not vectors - the resultant lies in the plane containing the two given vectors. Cross Product: - AxB ≠ BxA - cross products are vectors themselves - cross product vectors are perpendicular to the plane containing the two vectors.

What is the angle from the x-axis of a vector with components 2i -7j

Facts: x = 2; y = -7 tanɵ = opp (y)/adj (x) tanɵ = -7/2 tanɵ = -3.5 How do you get ɵ when you have tanɵ? tanɵ = -3.5 tanɵ/tan = -3.5/tan ɵ = -3.5/tan ɵ = 1/tan *(-3.5)....note that 1/tan is same as cot. ɵ = cot(-3.5) ɵ = -74.05° A° = -74.05

What are the components of a vector of 8 with an angle of 60 degrees from the x-axis?

Facts: |A| = 8 ɵ=60° A(subx) = 8cos(60°) = 4 A(suby) = 8sin(60°) = 6.93 A = 4i + 6.93j

Parallelogram Law of Vector Addition

Graphically, join vectors A and B at their tails, thus creating two sides of a parallelogram. To create the other two sides of the parallelogram, place the tail of B at the tip of A and the tail of A at the tip of B. Now you have a parallelogram. Where is the resultant? Well, draw a line from where the tails of A and B meet to where the tips of A and B meet. This line is your resultant. For two given vectors u and v enclosing an angle θ, the magnitude of the sum, |u + v|, is given by √(u2+v2+2uvcos(θ)).

Cross product and the cyclical order of vectors

Imagine a circle with points i→j→k and then back to i again. That means the cross product of: i x j = k; j x k = i; k x i = j It also means that if going in reverse order: j x i = (-k); k x j = (-i); i x k = (-j)

Vector addition (Triangle law)

Involves drawing individual vectors whereby the tip of one vector intersects the tail of the other. When adding the resultant, it starts from the unconnected tail of one vector to the unconnected tip of the other. For vector addition to work, vector lengths are drawn in proportion to their magnitudes and the angles between vectors must match the direction. This is called Scale Drawing.

Vector subtraction

It's as easy as it sounds. For example, subtracting vector A (2,-5) from vector B (3,3) yields (3-2,3-(-5)) or (1,8). The process of adding one vector to the negative value of another vector. To find the negative value of a vector, simply multiply the coordinates by -1. So (2,5) becomes (-2, -5).

Vector Addition using Algebra

Let us consider two vectors →A=a1^i+b1^j+c1^k and →B=a2^i+b2^j+c2^k →A + →B=(a1+a2)^i+(b1+b2)^j+(c1+c2)^k

Vector Subtraction using Algebra

Let us consider two vectors →A=a1^i+b1^j+c1^k and →B=a2^i+b2^j+c2^k →A - →B=(a1-a2)^i+(b1-b2)^j+(c1-c2)^k

Convert a vector with components 3i + 4j into a unit vector

Let's call the vector with components 3i+4J = v. We want to find the unit vector of v... called v^ v^ = v/|v|, where |v| = magnitude of v. Now, as v = 3i + 4j, we can rewrite as v=(3,4), where 3 and 4 are the x and y coordinates on a graph. As we know that the components of a vector are orthogonal to one another, we know we are talking about a right triangle with the hypotenuse being the unknown value...and the hypotenuse = magnitude = |v|. Using the pythagorean theorem, |v| = √((3^2)+4^2)) = 5 Plugging everything into the v^ = v/|v| formula, we have: v^ = 3i/5 + 4j/5 v^ = .6i + .8j

Magnitude of a vector

Magnitudes are ALWAYS real non-negative values that give both magnitude and direction. If the magnitude of a vector is not given, but its components are, you calculate the magnitude as follows: magnitude = |V| = √(i^2 + j^2 + k^2) Thus, vector A with components 3i + 4j has a magnitude of: |V| = √(3^2 + 4^2) |V| = √(9 + 16) |V| = √25 |V| = 5

Triangle inequality of vectors

Narration: difference of sides < 3rd side < sum of sides Math notation: (||a| - |b||) ≦ (|a-> + b->|) ≦ (|a| + |b|) Why the ≦? It's because sometimes two vectors will be in parallel... or they will be in antiparallel.

Properties of Vectors

The addition of vectors is commutative and associative. The additive identity of vectors is the zero vector. The additive inverse of a vector is the negative of the vector. The scalar multiplication of vectors is associative. The different properties of vectors are listed below: The addition of vectors is commutative and associative. A.B = B.A AxB≠BxA (yA)xB = ^i.^i=^j.^j=^k.^k=1 ^i.^j=^j.^k=^k.^i=0 ^i×^i=^j×^j=^k×^k=0 ^i×^j=^k ; ^j×^k=^i ; ^k×^i=^j ^j×^i=−^k ; ^k×^j=−^i ; ^i×^k=−^j

Vector components

They look like graph coordinates. In two dimensional space, they may look like (0,0). In 3-dimensional space, they may look like (0,0,0). A vector in n-dimensional space is represented by n component values. These values correspond to the distance along each axis that vector travels based on its magnitude and direction. The x,y,z values common to traditional algebra are referenced as i,j,k in vector algebra.

When are two vectors equal?

Two vectors are equal if the relative components of each vector are equal A→ and B→ are equal if: a1=a2; b1=b2: c1=c2.

Vector notation

Vector "A" is denoted as A-> (the -> is on top of the A) Standard form of a vector with its components is: A-> = ai + bj + ck, where a,b,c are numeric scalar values and i,j,k are the unit vectors along the x,y,z axis respectively. *i,j,k - are vectors in and of themselves (having magnitude and direction) and are orthogonal (90 degree angles) to one another. You might read a vector described as: 1) 4 newtons east and 3 newtons north 2) 4i and 3j 3) (4,3) x,y, and z correspond to i, j, and k in vector notation.

Cross product of vectors

a → x b→ = 2 options: 1) Matrix math (see matrix below) i j k a1 b1 c1 a2 b2 c2 a → x b→ = i(b1c2-b2c1) - j(a1c2-a2c1) + k(a1b2-a2b1) 2)a → x b→ = |a|*|b|*sin(θ)n *n is the symbol representing the vector

What are the x and y coordinates of a vector with magnitude 5 and angle 126.87?

x coordinate = cosɵ y coordinate = sinɵ x = 5cos(126.87) = -3 y = 5sin(126.87) = 4 (-3,4) are the coordinates of a vector with a magnitude of 5 at 126.87 degrees.