Calculus III Project I

answers.txt

==> question2a.txt <==
Distance: 89/sqrt(1635)

==> question2b.txt <==
Distance: 40/sqrt(104)

==> question2c.txt <==
Distance: 64/sqrt(222)

==> question3.txt <==
Equation: -5(x - 3) + -10(y + 1) + 5(z - 2) = 0

==> question4.txt <==
Equation: -5(x - 4) + -10(y - 0) + 5(z - 3) = 0

==> question5.txt <==
Equation: -5(x - 5) + -10(y - 1) + 5(z - 4) = 0

==> question6.txt <==
Equation: -5(x - 6) + -10(y - 2) + 5(z - 5) = 0

==> question7.txt <==
Equation: -5(x - 7) + -10(y - 3) + 5(z - 6) = 0

==> question8.txt <==
Area: sqrt(150)

==> question9.txt <==
Area: sqrt(150)

==> question10.txt <==
Area: sqrt(150)

==> question11.txt <==
Area: sqrt(150)

==> question12.txt <==
Area: sqrt(150)

parallelogram.js

function vector(from, to) {
	return {
		x: to.x - from.x,
		y: to.y - from.y,
		z: to.z - from.z,
	};
}

function crossProduct(u, v) {
	return {
		x: u.y * v.z - v.y * u.z,
		y: -(u.x * v.z - v.x * u.z),
		z: u.x * v.y - v.x * u.y,
	};
}

function signed(num) {
	if (num > 0) {
		return "+ " + num;
	} else {
		return "- " + (-num).toString();
	}
}

function mag2(vec) {
	return vec.x * vec.x + vec.y * vec.y + vec.z * vec.z;
}

// // question 8
// const a = {x:  3, y: -1, z:  2};
// const b = {x:  2, y:  1, z:  5};
// const c = {x:  1, y: -2, z: -2};

// // question 9
// const a = {x:  4, y:  0, z:  3};
// const b = {x:  3, y:  2, z:  6};
// const c = {x:  2, y: -1, z: -1};

// // question 10
// const a = {x:  5, y:  1, z:  4};
// const b = {x:  4, y:  3, z:  7};
// const c = {x:  3, y:  0, z:  0};

// // question 11
// const a = {x:  6, y:  2, z:  5};
// const b = {x:  5, y:  4, z:  8};
// const c = {x:  4, y:  1, z:  1};

// question 12
const a = {x:  7, y:  3, z:  6};
const b = {x:  6, y:  5, z:  9};
const c = {x:  5, y:  2, z:  2};

const AB = vector(a, b);
const AC = vector(a, c);

const n = crossProduct(AB, AC);

console.log(`Area: sqrt(${mag2(n)})`);

plane-points.js

function vector(from, to) {
	return {
		x: to.x - from.x,
		y: to.y - from.y,
		z: to.z - from.z,
	};
}

function crossProduct(u, v) {
	return {
		x: u.y * v.z - v.y * u.z,
		y: -(u.x * v.z - v.x * u.z),
		z: u.x * v.y - v.x * u.y,
	};
}

function signed(num) {
	if (num > 0) {
		return "+ " + num;
	} else {
		return "- " + (-num).toString();
	}
}

// question 3
const a = {x:  3, y: -1, z:  2};
const b = {x:  2, y:  1, z:  5};
const c = {x:  1, y: -2, z: -2};

// question 4
// const a = {x:  4, y:  0, z:  3};
// const b = {x:  3, y:  2, z:  6};
// const c = {x:  2, y: -1, z: -1};

// question 5
// const a = {x:  5, y:  1, z:  4};
// const b = {x:  4, y:  3, z:  7};
// const c = {x:  3, y:  0, z:  0};

// question 6
// const a = {x:  6, y:  2, z:  5};
// const b = {x:  5, y:  4, z:  8};
// const c = {x:  4, y:  1, z:  1};

// question 7
// const a = {x:  7, y:  3, z:  6};
// const b = {x:  6, y:  5, z:  9};
// const c = {x:  5, y:  2, z:  2};

const AB = vector(a, b);
const AC = vector(a, c);

const n = crossProduct(AB, AC);

console.log(`Equation: ${n.x}(x ${signed(-a.x)}) + ${n.y}(y ${signed(-a.y)}) + ${n.z}(z ${signed(-a.z)}) = 0`);

skew-lines.js

function crossProduct(u, v) {
	return {
		x: u.y * v.z - v.y * u.z,
		y: -(u.x * v.z - v.x * u.z),
		z: u.x * v.y - v.x * u.y,
	};
}

function dotProduct(u, v) {
	return u.x * v.x + u.y * v.y + u.z * v.z;
}

function mag2(vec) {
	return vec.x * vec.x + vec.y * vec.y + vec.z * vec.z;
}

function vector(from, to) {
	return {
		x: to.x - from.x,
		y: to.y - from.y,
		z: to.z - from.z,
	};
}

function abs(num) {
	if (num > 0) {
		return num;
	} else {
		return -num;
	}
}

// question 2.a
const l1 = {x:  5, y:  5, z: -4};
const l2 = {x:  1, y:  8, z: -3};

const p = {x:  4, y:  5, z:  1};
const q = {x:  4, y: -6, z:  7};

// question 2.b
// const l1 = {x:  2, y:  4, z:  6};
// const l2 = {x: -1, y:  1, z:  1};
// 
// const p = {x:  0, y:  0, z:  0};
// const q = {x:  1, y:  4, z: -1};

// // question 2.c
// const l1 = {x:  3, y: -1, z:  1};
// const l2 = {x:  4, y:  1, z: -3};
// 
// const p = {x:  0, y:  2, z: -1};
// const q = {x:  1, y: -2, z: -3};

const pq = vector(p, q);

const n = crossProduct(l1, l2);

// project pq onto n then take magnitude
console.log(`Distance: ${abs(dotProduct(n, pq))}/sqrt(${mag2(n)})`);