Question

asked 2021-06-06

Let vectors \(A'=(2,\ 1,\ -4)\)

\(B'=(-3,\ 0,\ 1)\)

and \(C'=(-1,\ -1,\ 2)\)

Calculate the following:

\(2(B'\cdot3C')=?\)

\(B'=(-3,\ 0,\ 1)\)

and \(C'=(-1,\ -1,\ 2)\)

Calculate the following:

\(2(B'\cdot3C')=?\)

asked 2021-06-10

Find an equation for the plane containing the two (parallel) lines

\(v_1=(0,1,-2)+t(2,3,-1)\)

and \(v_2=(2,-1,0)+t(2,3,-1).\)

\(v_1=(0,1,-2)+t(2,3,-1)\)

and \(v_2=(2,-1,0)+t(2,3,-1).\)

asked 2021-05-16

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

A. Let y=f(x) be the equation of C. Find f(x).

B. Find the slope at P of the tangent to C.

C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?

D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.

E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.

Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.

asked 2021-05-29

Which of the following expressions are meaningful? Which are meaningless? Explain.

a) \((a\cdot b)\cdot c\)

\((a\cdot b)\cdot c\) has ? because it is the dot product of ?.

b) \((a\cdot b)c\)

\((a\cdot b)c\) has ? because it is a scalar multiple of ?.

c) \(|a|(b\cdot c)\)

\(|a|(b\cdot c)\) has ? because it is the product of ?.

d) \(a\cdot(b+c)\)

\(a\cdot(b+c)\) has ? because it is the dot product of ?.

e) \(a\cdot b+c\)

\(a\cdot b+c\) has ? because it is the sum of ?.

f) \(|a|\cdot(b+c)\)

\(|a|\cdot(b+c)\) has ? because it is the dot product of ?.

a) \((a\cdot b)\cdot c\)

\((a\cdot b)\cdot c\) has ? because it is the dot product of ?.

b) \((a\cdot b)c\)

\((a\cdot b)c\) has ? because it is a scalar multiple of ?.

c) \(|a|(b\cdot c)\)

\(|a|(b\cdot c)\) has ? because it is the product of ?.

d) \(a\cdot(b+c)\)

\(a\cdot(b+c)\) has ? because it is the dot product of ?.

e) \(a\cdot b+c\)

\(a\cdot b+c\) has ? because it is the sum of ?.

f) \(|a|\cdot(b+c)\)

\(|a|\cdot(b+c)\) has ? because it is the dot product of ?.