Vectors

Multiplying a vector by a scalar

- A scalar is just a constant (has no direction) - If the scalar is positive, the vector's magnitude changes by that number - If the scalar is negative, the vector's direction becomes the opposite and its magnitude changes by the absolute value of the number

Drawing vectors on a graph

- The initial and terminal points with a line the correct magnitude between them, and an arrow at the end that signifies which direction the vector is going - The arrow is CRUCIAL in a drawing

Vector notation

- The starting point is called the initial point - The ending point is called the terminal point - If A is initial and B is terminal, the vector can be notated AB with a line with an arrow over it - Vectors can also be called by one letter, typically from the end of the alphabet (such as u with an arrow line over it) - In textbooks, the one letter vectors may also be bolded

Subtracting vectors

- To subtract vectors, you have to add the opposite of the vector - The opposite vector is the vector of the same magnitude in the opposite direction--this means if the vector is going up and right, its opposite would be going down and left - Connect them tip to tail in the same way, just with the opposite vector --> the resultant vector is the product of the subtraction - Vector subtraction is NOT COMMUTATIVE

How to find vector magnitude

- slide the vector to the origin - Construct a right triangle and solve for the hypotenuse using pythag - The length of the hypotenuse is the magnitude of the vector

What is the name of the resultant vector when the addition/subtraction completely cancels out?

It is called a zero vector

vector

directed line segment that has magnitude (length) and direction (θ)

Unit vectors

i (with a hat): unit vector in the horizontal direction <1, 0> j (with a hat): unit vector in the vertical direction <0, 1>

Vector operation in 3D

i hat = right/left j hat = up/down k hat = forward/backward - To find the magnitude of a 3D vector, do pythag but with all three components - Dot product in 3D is done the same way as dot product in 2D --> can still use the shortcut

When are 2 vectors considered equal?

if they have the same magnitude and direction

What does orthogonal mean

perpendicular, but concerning vectors

Notation for vector magnitude

| AB | = magnitude of vector AB - It is just absolute value bars

Ramp problem

µ = coefficient of static friction (will be given in the problem) Ffric = µ • Fnormal - Draw the perpendicular and parallel components of the object, then connect them (forming another right triangle) - Label the sides of the right triangle Minimum force to move the object down the ramp: Ffric - parallel force Minimum force to move the object up the ramp: Ffric + parallel force Force to keep the object from moving: parallel force

How to find vector direction (the angle)

- After sliding the vector to the origin, the angle in standard position is the vector direction - Because you have constructed a right triangle to find the magnitude, use tan inverse to find the angle (or if its an obvious ratio, use the special right triangles)

How do you check if the resultant vector from cross product is actually perpendicular to the original two?

- Do the dot product with the resultant and each of the original vectors - The result should be 0 in both cases - This is because if it is perpendicular, meaning at a 90 degree angle, then cos90 = 0

Finding vector components given magnitude and direction

- Draw a picture (quite similar to right triangle trig w the unit circle!) - The magnitude is the length of the hypotenuse --> drop an altitude to form a right triangle - You know which quadrant the triangle falls in due to the direction angle (find the reference angle if needed) <V₁, V₂> can be calculated as follows: V₁ = |v|cosθ, V₂ = |v|sinθ where θ is the reference angle found from the direction angle *make sure to figure out if positive or negative depending on the quadrant!*

Adding vectors

- Draw both vectors on a coordinate plane - Connect the vectors tip to tail, then draw the vector that goes from the tail of the first to the tip of the last - The connecting vector is the resultant vector - The resultant vector has its own magnitude and direction - Vector addition is COMMUTATIVE

Finding vector magnitude and direction angle given components

- Draw the vector in the correct quadrant using the given components - Find the hypotenuse with pythag - The magnitude is the length of the hypotenuse - Use tan inverse the find the reference angle - Make sure that the direction angle takes into account the quadrant (may have to add/subtract the reference angle from pi or 2pi)

Finding a unit vector in the same direction as another vector

- First draw the the normal vector, and drop an altitude, create a right triangle, and label the sides - Because unit vectors have a magnitude of 1, scale all the sides down by dividing everything by the hypotenuse (just like the unit circle!) - The unit vector in the same direction has the components of this new triangle (and a magnitude of 1)

How can you algebraically determine the components of a resultant vector?

- First, write each vector in terms of i and j components (4i + 3j, for example --> do NOT use < > with unit vectors!!!) - Then, add/subtract as necessary - The resulting i/j components represent the resultant vector

Vector components

- How far a vector goes horizontally and vertically - These are basically the coordinate points, but because they AREN'T points, you have to notate them differently u = <3, 3> - The x and y distances (same format as coordinate points)

Object hanging vector application problems: strategy

- If an object is hanging, ask yourself: what needs to be true for it to not fall and not swing back and forth? - The upwards force must counteract the downwards force - The left and right forces must be equal - Knowing this, see what can be set equal to each other to form a system of equations where you may need to solve for two or more variables - AND form right triangles wherever possible and write those components down!

Cross product

- Only happens in 3D, and yields a vector that is perpendicular to the original 2 vectors!!!!!! - |a x b| = |a||b|sinθ - No shortcut for this unfortunately - Use the right hand rule to find direction (i, j, k or -i, -j, -k) - Multiply the magnitudes to find the magnitude of the resultant vector - You have the expand this out and find the cross product of each individual i, j, or k - i and i, j and j, k and k cross products all result in 0 vector (because the angle between them is 0 and the sin of 0 is 0

Dot product

- Results in a scalar - Vectors must be tail to tail to find the dot product - a • b = |a||b|cosθ Shortcut: multiply the i coefficients together and the j coefficients together, then add them - To find the angle between 2D vectors, first find the dot product, then plug that in along with the magnitudes of the two vectors, then find the angle using inverse cosine

Vector application problems

- Write every vector in terms of its horizontal and vertical components (i and j) - Add them all up (meaning add all the horizontals together to get the total i, and all the verticals together to get the total j) - These can be turned in to the bracketed vector components if easier - Find the magnitude of the resultant vector using pythag - Find the direction angle of the resultant vector using tan inverse (keep positive, adjust based on quadrant) - Answer what the question is asking in a complete sentence!

Writing vector components in terms of i and j

-3i + 5j = <-3, 5> i is the x direction j is the y direction