Vector Operations
If vector a = 4i + 2j - k and vector b = -3i + 6j, what is 2a + b?
5i + 10j - 2k
Find the derivative of:
y(x) = 6e-2x
y'(x) = -12e-2x
If f(x,y) = 6x2y + 5xy2, what are the partial derivatives with respect to x and y?
WRT x = 12xy + 5y2
WRT y = 6x2 + 10xy
Add vector a = 6i + 8j and b = 5i + 3j + 6k together. What is the magnitude of the resultant vector?
a + b = 11i + 11j + 6k, |a+b| = 16.67
Differentiate f(x) = 1/x2 - 4/x5
-2/x3 + 20/x6
Find the general solution of u(x,y) that satisfies the following two partial differential equations:
1. d/dx (u(x,y)) = 7 cos (y) + 14x
2. d/dy (u(x,y)) = -7x sin (y) - 5
u(x,y) = 7x cos (y) + 7x2 - 5y + C
If a = 2i + 6j and b = -4i + 8j, is a.b a scalar or a vector? What is a.b?
a.b is a scalar, the value is 40
What rule do we need to use, and what is the derivative (rate of change) of y = (5x−2)3 ?
We need to use chain rule (function inside of a function), the answer is 15(5x-2)2
If f(x,y,z,p) = 6x2y4z3p3, the partial derivatives with respect to x,y,z and p are...
WRT x = 12xy4z3p3
WRT y = 24x2y3z3p3
WRT z = 18x2y4z2p3
WRT p = 18x2y4z3p2
If |a| = 5 and |b| = (3/7), and the angle between these two vectors is pi/12, what is a.b equal to?
2.07
Differentiate f(x) = x sin x
If f(x,y,z) = 3ex + z2 - 6xy + 2y6, then the gradient vector is...
(3ex - 6y)i + (-6x + 12y5)j + (2z)k
a = 6i + 2j - 3k and b = -4i + j - 3k, a x b is...
-3i + 30j + 14k
What rule do we need to use, and what is the derivative for the following function:
f(x) = 3cos(2x)sin(x)
We need to use the product rule (used when there are two functions multiplied together), and the answer is:
3(-5cos(x)+6cos3(x))
If f(x,y) = sin(x3y), then the gradient vector ∇ f(x,y) is...
[cos (x3y) * 3x2y] i + [cos (x3y) * x3] j