University:
Amherst College
Course:
MATH 211 | Multivariable Calculus
Academic year:

2024
Views:

135

Pages:

4
Author:

Bunicco14j

1. Elementary Vector Analysis Vectors, Points, and Position Vectors: The position vector of a point P is the vector from the origin to P . For example, in R3 , a vector ~v = hx, y, zi is the position vector of the point P = (x, y, z). In this way, we often identify points as vectors and vice versa. Parametrized Curves Parametrized Curves: Given a continuous vector function ~r(t) = hx(t), y(t)i or ~r(t) = hx(t), y(t), z(t)i, the set of points with position vector ~r(t) for some t defines a curve C. The components x(t), y(t), and z(t) are called parametric equations for the parametrized curve C. The variable t is called the parameter. The tangent vector to a curve at the point P = ~r(t) is r~0 (t) = hx0 (t), y 0 (t)i or r~0 (t) = hx0 (t), y 0 (t), z 0 (t)i. The line through the point P = ~r(t) parallel to the vector r~0 (t) is called the tangent line to C at P . Ex. 1. (a) Sketch the curve C with parametric equations x(t) = t + 1, y(t) = t2 2t, 1  t  3. (b) Show that C is an arc of the parabola y = x2 4x + 3 and use this fact to give another parametrization of C. t x y -1 0 3 0 1 0 1 2 3 Ex. 2. What curve is represented by the parametric equations? (a) x(t) = 2 cos(t), y(t) = 2 sin(t), 0  t  2⇡ (b) x(t) = sin(2t), y(t) = cos(2t), 0  t  2⇡ (c) Compute the tangent vector to the curve ~r(t) = h2 cos(t), 2 sin(t)i and show that it is perpendicular to the curve at any point (x(t), y(t)). Lines in R2 and R3 Vector Equation: ~r(t) = r~0 + t~v Parametric Equations: x(t) = x0 + at, y(t) = y0 + bt, z(t) = z0 + ct ~r(t) = hx(t), y(t), z(t)i are the position vectors of the points (x(t), y(t), z(t)) on the line r~0 = hx0 , y0 , z0 i is the position vector of a fixed point P0 = (x0 , y0 , z0 ) on the line ~v = ha, b, ci is a direction vector of the line, i.e. ~v is any vector parallel to the line Rmk. For lines in R2 , just delete all of the third coordinates above. Ex. 3. Find parametric equations for the line that passes through the points (2, 0, 1) and ( 1, 1, 1). Ex. 4. Find parametric equations for the line segment from (3, 1, 2) to ( 5, 4, 1). Ex. 5. Find parametric equations for the line that passes through the point (2, 0, 1) and is parallel to the line given by parametric equations x(t) = 3t, y(t) = 5, and z(t) = t + 4. Ex. 6. Find parametric equations for the tangent line to the parametrized curve x(t) = t + 1, y(t) = t2 at the point (0, 3). 2t, Planes in R3 Vector Equation: ~n · (~r r~0 ) = 0 Scalar Equations: a(x x0 ) + b(y y0 ) + c(z z0 ) = 0 or ax + by + cz = d ~r = hx, y, zi are the position vectors of points (x, y, z) in the plane r~0 = hx0 , y0 , z0 i is the position vector of a fixed point P0 = (x0 , y0 , z0 ) in the plane ~n = ha, b, ci is the normal vector, i.e. any vector orthogonal to the plane Ex. 7. Find an equation of the plane that passes through the point (1, 1, 0) and is perpendicular to the line with parametric equations x(t) = 2 + 3t, y(t) = 1 t, z(t) = 2 + 2t. Ex. 8. Find an equation of the plane that passes through the points (1, 0, 2), (4, 2, 3), and (2, 0, 4). Ex. 9. Find an equation of the plane that passes through the point (1, 0, 2) and contains the line x = 2t + 2, y = 2t, z = 4 t. Ex. 10. Find parametric equations for the line of intersection for the two planes 2x + y + z = 8 and x + 2y z = 1.

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