Vector Equation of the line
What is the formula of vector equation of the line?
r = a + tb
You are running in a race. You overtake the person in second place. Where do you rank in the race now?
Second
The angle between two vectors measured in which range?
0 ≤ x ≤ 180
How do we call quantities which have only magnitude?
scalars
Given vectors a = (2, -1) and b = (-3, 5), find the dot product of a and b.
The dot product of vectors a and b is a ⋅ b = (2 * (-3)) + (-1 * 5) = -6 - 5 = -11.
What is the formula for parametric equation of the line?
x = a1 + tb1; y = a2 + tb2
There are five sisters in a room. Anne is reading a book, Margaret is cooking, Kate is playing chess, and Marie is doing the laundry. What's the fifth sister doing?
Playing chess with Kate
Formula of the angle between two lines
cos θ = (b1⋅b2)/(|b1||b2|)
How do we find vector between two points?
we substract initial point from the final point, x and y values seperately (x2-x1; y2-y1)
Given vector a = (3, -2) and vector b = (-1, 5), find the magnitude of the vector sum a + b.
To find the magnitude of the vector sum a + b, we first need to calculate the sum of the two vectors: a + b = (3 - 1, -2 + 5) = (2, 3). Then, we find the magnitude of this resultant vector: |a + b| = √(2² + 3²) = √13.
What is the formula for cartesian equation of the line?
t = (x-a1)/b1 = (y-a2)/b2
How much was $1.00 USD worth in 1976?
$1.00 USD
Free Points
Easy 300 points
What is the dot product of two vectors?
The scalar product of two vectors, equal to the product of their magnitudes multiplied by the cosine of the angle between them.
If u = –i + 3j, v = 7i – 4j and w = 2i + j then find
(3u) · (v + w).
3u = –3i + 9j
v + w = 9i – 3j
(3u) · (v + w) = -27 +(-27) =-54
what does T stand for in the equation and what is it called?
t - parameter
it shows length and direction
A man, his wife, and their son are in a car accident. They are all rushed to the hospital and the doctor says, “I can’t operate on him, he’s my son.” Why?
The doctor is the man’s father and the boy’s grandfather.
Given two non-collinear vectors u = (2, 3) and v = (5, -1) in a 2D plane, what are the angles between each vector and the x-axis, rounded to the nearest degree?
The angle between vector u and the x-axis is approximately 56 degrees, and the angle between vector v and the x-axis is approximately 345 degrees (or equivalently, -15 degrees).
Using the magnitude formula, find the magnitude of the vector with u = (2, 5)?
Given:
Vector u = (2,5)
Using magnitude formula,
|u| = √(x^2 + y^2) = √(22 + 52) = √(4 + 25) = √29 ≈ 5.385
Find the angle between two vectors a = {7; 1} and b = {5; 5}.
calculate dot product of vectors:
a·b = 5 · 7 + 1 · 5 = 35 + 5 = 40.
Calculate vectors magnitude:
|a| = √72 + 12 = √49 + 1 = √50 = 5√2
|b| = √52 + 52 = √25 + 25 = √50 = 5√2
Calculate the angle between vectors:
cos α = (a · b)/ (|a|· |b| ) = 40/5√2x5√2 = 0.8
a = 37 degrees
How would you find the intersection point between two lines represented by their vector equations?
To find the intersection point between two lines represented by their vector equations, set the equations equal to each other and solve the resulting system of equations for the variables representing the coordinates of the intersection point
T = 7777
R = 1111
Y = 5555
N = 4444
E = ?
E = 3333
T = 7+7+7+7 = TWENTY EIFH(T)
R = 1+1+1+1 = FOU(R)
Y = 5+5+5+5 = TWENT(Y)
N = 4+4+4+4 = SIXTEE(N)
3+3+3+3 = TWELV(E)
Find the angle between two vectors a = {3; 4} and b = {4; 3}.
Calculate dot product of vectors:
a·b = 3 · 4 + 4 · 3 = 12 + 12 = 24.
Calculate vectors magnitude:
|a| = √32 + 42 = √9 + 16 = √25 = 5
|b| = √42 + 32 = √16 + 9 = √25 = 5
Calculate the angle between vectors:
cos α = (a · b)/(|a|·|b|) = 24/25 = 0.96
a = 16
Length of the given vector
The endpoints of the given vector are OA = <-3, -1> = <x1, y1> and OB = <2, -5> = <x2, y2>
|A| = √((x2 - x1)^2 + (y2 - y1)^2)
= √ [(2 - (-3))^2 + (-5-(-1))^2]
= √(5^2 + (-4)^2)
= √(25 + 16)
= √41 = 6.4
= 3√5
Two forces of magnitude 8 N and 12 N act on an object at angles of 30 degrees and 60 degrees counterclockwise from the positive x-axis, respectively. Determine the magnitude and direction of the resultant force acting on the object.
The x-component of the resultant force is 8cos(30°) + 12cos(60°) = 4√3 + 6 N, and the y-component is 8sin(30°) + 12sin(60°) = 4 + 6√3 N. Using these components, the magnitude of the resultant force is √((4√3 + 6)² + (4 + 6√3)²) ≈ 16.85 N. The direction is given by the angle θ such that tan(θ) = (4 + 6√3) / (4√3 + 6), so θ ≈ 45.71°.