MA 125 Exam 1
Feihong Wang
TOTAL POINTS
QUESTION 1
other than $$\frac{1}{\|\vec v_1\|}$$ or $$-
\frac{1}{\|\vec v_1\|}$$
1 Q1
ÊÊ + 0 pts No scalar multiple of $$\vec v_1$$ is
✓ + 10 pts Everything correct and work shown
given
ÊÊ + 4 pts Part (a) correct and work shown
ÊÊ + 6 pts Part (b) correct and work shown Part (b): formulas applied (explicitly, or
ÊÊ + 0 pts Everything blank or totally incorrect implicitly through computation without
general formulas given)
Part (a): $$\|\vec v_1\|$$
ÊÊ + 3.5 pts Correct and relevant formulas for
ÊÊ + 2 pts $$\|\vec v_1\|$$ computed correctly
projection and dot product are used, i.e.
ÊÊ + 1.5 pts $$\|\vec v_1\|$$ computed with
$$\textrm{Proj}_{\vec v_1}\left(\vec
arithmetic mistake
v_2\right)=\left(\dfrac{\|\vec v_1\|\|\vec
ÊÊ + 1 pts $$\|\vec v_1\|$$ computed with mistake
v_2\|\cos\theta}{\|\vec v_1\|^2}\right) \vec v_1$$
in formula
or equivalent
ÊÊ + 0.5 pts $$\|\vec v_1\|$$ referenced, but no
ÊÊ + 2.5 pts Minor mistake in formulas for
attempt to compute
projection and dot product, but appropriate
ÊÊ + 0 pts No reference to $$\|\vec v_1\|$$
forms clearly intended (e.g. "geometric" formula
Part (a): scaling $$\vec v_1$$
for dot product as opposed to component-wise)
ÊÊ + 2 pts $$\vec v_1$$ scaled correctly by $$\pm
ÊÊ + 2 pts Only one of either the correct projection
\frac{1}{\|\vec v_1\|}$$ (regardless of magnitude
or geometric dot product formulas used (with
being computed correctly)
possibly a significant mistake in the attempted
ÊÊ + 1.5 pts $$\vec v_1$$ scaled by $$\pm
application of the other formula)
\frac{1}{\|\vec v_1\|}$$, but with arithmetic
ÊÊ + 1 pts Only one of either the projection or
mistake or left with unsimplified scalar factor
geometric dot product formulas used, with
ÊÊ + 1 pts $$\vec v_1$$ scaled by only either
mistake in formulation
$$\frac{1}{\|\vec v_1\|}$$ or $$-\frac{1}{\|\vec
ÊÊ + 0 pts No identifiable use of relevant formula
v_1\|}$$ (along with possibly a second vector
Part (b): evaluation
from scaling by some other incorrect factor)
ÊÊ + 2.5 pts Projection (including dot product)
ÊÊ + 0.5 pts $$\vec v_1$$ is scaled by something
, formula(s) which are either correct or with minor $$\Pi_2$$ found completely incorrectly, or not at
mistake, are evaluated correctly (relative to all
computation of $$\|\vec v_1\|$$ from part (a))
Normal vector for $$\Pi_3$$
ÊÊ + 2 pts Formula(s) evaluated with arithmetic
ÊÊ + 4 pts Cross product computed (or system of
mistake and/or incompletely (this includes leaving
equations solved) correctly from normal vectors
$$\cos\left(\frac{2\pi}{3}\right)$$ unsimplified but
found for $$\Pi_1,\Pi_2$$
NOT things like leaving a fraction unsimplified)
ÊÊ + 3.5 pts Cross product computed (or system of
but with only minor steps left
equations solved) from normal vectors found for
ÊÊ + 1.5 pts Formula(s) evaluated incompletely,
$$\Pi_1,\Pi_2$$, but with arithmetic mistake
with terms left unsubstituted (e.g. angle left as
ÊÊ + 3 pts Cross product computed (or system of
just unknown $$\theta$$)
equations solved) from normal vectors found for
ÊÊ + 1.5 pts Formula(s) evaluated with algebraic
$$\Pi_1,\Pi_2$$, but with minor mistake in cross
mistake, e.g. incorrect simplification of exponents
product formula or minor algebraic mistake
ÊÊ + 1 pts Formula(s) evaluated with mistake in
ÊÊ + 2 pts Attempt to compute cross product (or
vector operations, i.e. using
solve system of equations) from normal vectors
addition/scaling/products/"division" in an
found for $$\Pi_1,\Pi_2$$, but with major mistake
incorrect way beyond arithmetic/algebraic
in cross product formula or major algebraic
mistakes
mistake
ÊÊ + 1 pts Some minor steps taken correctly in
ÊÊ + 1.5 pts Cross product or system of equations
evaluation, but with significant computation left
setup correctly, relative to normal vectors of
incomplete (this includes the case of incomplete
$$\Pi_1,\Pi_2$$, but no attempt to compute
formulas set up, e.g. only dot product)
ÊÊ + 1 pts Some reference to cross product, or
ÊÊ + 0 pts No correct steps are taken to evaluate
multiple dot products equaling zero, but not set
formulas stated
up (with normal vectors of $$\Pi_1,\Pi_2$$)
QUESTION 2 ÊÊ + 0 pts No relevant steps towards finding
normal vector of $$\Pi_3$$
2 Q2
✓ + 8 pts Everything correct and work shown Equation of plane $$\Pi_3$$
ÊÊ + 2 pts Equation of plane correctly given,
Normal vectors for $$\Pi_1,\Pi_2$$
relative to normal vector found
ÊÊ + 2 pts Normal vectors for $$\Pi_1$$ and
ÊÊ + 1.5 pts Correct form of equation used, but
$$\Pi_2$$ found correctly
mistake in substituting values (for components of
ÊÊ + 1 pts Normal vectors for $$\Pi_1$$ and
normal vector, coordinates of point)
$$\Pi_2$$ found with minor mistake(s)
ÊÊ + 1 pts Correct general form of equation with
ÊÊ + 0 pts Normal vectors for $$\Pi_1$$ and
Feihong Wang
TOTAL POINTS
QUESTION 1
other than $$\frac{1}{\|\vec v_1\|}$$ or $$-
\frac{1}{\|\vec v_1\|}$$
1 Q1
ÊÊ + 0 pts No scalar multiple of $$\vec v_1$$ is
✓ + 10 pts Everything correct and work shown
given
ÊÊ + 4 pts Part (a) correct and work shown
ÊÊ + 6 pts Part (b) correct and work shown Part (b): formulas applied (explicitly, or
ÊÊ + 0 pts Everything blank or totally incorrect implicitly through computation without
general formulas given)
Part (a): $$\|\vec v_1\|$$
ÊÊ + 3.5 pts Correct and relevant formulas for
ÊÊ + 2 pts $$\|\vec v_1\|$$ computed correctly
projection and dot product are used, i.e.
ÊÊ + 1.5 pts $$\|\vec v_1\|$$ computed with
$$\textrm{Proj}_{\vec v_1}\left(\vec
arithmetic mistake
v_2\right)=\left(\dfrac{\|\vec v_1\|\|\vec
ÊÊ + 1 pts $$\|\vec v_1\|$$ computed with mistake
v_2\|\cos\theta}{\|\vec v_1\|^2}\right) \vec v_1$$
in formula
or equivalent
ÊÊ + 0.5 pts $$\|\vec v_1\|$$ referenced, but no
ÊÊ + 2.5 pts Minor mistake in formulas for
attempt to compute
projection and dot product, but appropriate
ÊÊ + 0 pts No reference to $$\|\vec v_1\|$$
forms clearly intended (e.g. "geometric" formula
Part (a): scaling $$\vec v_1$$
for dot product as opposed to component-wise)
ÊÊ + 2 pts $$\vec v_1$$ scaled correctly by $$\pm
ÊÊ + 2 pts Only one of either the correct projection
\frac{1}{\|\vec v_1\|}$$ (regardless of magnitude
or geometric dot product formulas used (with
being computed correctly)
possibly a significant mistake in the attempted
ÊÊ + 1.5 pts $$\vec v_1$$ scaled by $$\pm
application of the other formula)
\frac{1}{\|\vec v_1\|}$$, but with arithmetic
ÊÊ + 1 pts Only one of either the projection or
mistake or left with unsimplified scalar factor
geometric dot product formulas used, with
ÊÊ + 1 pts $$\vec v_1$$ scaled by only either
mistake in formulation
$$\frac{1}{\|\vec v_1\|}$$ or $$-\frac{1}{\|\vec
ÊÊ + 0 pts No identifiable use of relevant formula
v_1\|}$$ (along with possibly a second vector
Part (b): evaluation
from scaling by some other incorrect factor)
ÊÊ + 2.5 pts Projection (including dot product)
ÊÊ + 0.5 pts $$\vec v_1$$ is scaled by something
, formula(s) which are either correct or with minor $$\Pi_2$$ found completely incorrectly, or not at
mistake, are evaluated correctly (relative to all
computation of $$\|\vec v_1\|$$ from part (a))
Normal vector for $$\Pi_3$$
ÊÊ + 2 pts Formula(s) evaluated with arithmetic
ÊÊ + 4 pts Cross product computed (or system of
mistake and/or incompletely (this includes leaving
equations solved) correctly from normal vectors
$$\cos\left(\frac{2\pi}{3}\right)$$ unsimplified but
found for $$\Pi_1,\Pi_2$$
NOT things like leaving a fraction unsimplified)
ÊÊ + 3.5 pts Cross product computed (or system of
but with only minor steps left
equations solved) from normal vectors found for
ÊÊ + 1.5 pts Formula(s) evaluated incompletely,
$$\Pi_1,\Pi_2$$, but with arithmetic mistake
with terms left unsubstituted (e.g. angle left as
ÊÊ + 3 pts Cross product computed (or system of
just unknown $$\theta$$)
equations solved) from normal vectors found for
ÊÊ + 1.5 pts Formula(s) evaluated with algebraic
$$\Pi_1,\Pi_2$$, but with minor mistake in cross
mistake, e.g. incorrect simplification of exponents
product formula or minor algebraic mistake
ÊÊ + 1 pts Formula(s) evaluated with mistake in
ÊÊ + 2 pts Attempt to compute cross product (or
vector operations, i.e. using
solve system of equations) from normal vectors
addition/scaling/products/"division" in an
found for $$\Pi_1,\Pi_2$$, but with major mistake
incorrect way beyond arithmetic/algebraic
in cross product formula or major algebraic
mistakes
mistake
ÊÊ + 1 pts Some minor steps taken correctly in
ÊÊ + 1.5 pts Cross product or system of equations
evaluation, but with significant computation left
setup correctly, relative to normal vectors of
incomplete (this includes the case of incomplete
$$\Pi_1,\Pi_2$$, but no attempt to compute
formulas set up, e.g. only dot product)
ÊÊ + 1 pts Some reference to cross product, or
ÊÊ + 0 pts No correct steps are taken to evaluate
multiple dot products equaling zero, but not set
formulas stated
up (with normal vectors of $$\Pi_1,\Pi_2$$)
QUESTION 2 ÊÊ + 0 pts No relevant steps towards finding
normal vector of $$\Pi_3$$
2 Q2
✓ + 8 pts Everything correct and work shown Equation of plane $$\Pi_3$$
ÊÊ + 2 pts Equation of plane correctly given,
Normal vectors for $$\Pi_1,\Pi_2$$
relative to normal vector found
ÊÊ + 2 pts Normal vectors for $$\Pi_1$$ and
ÊÊ + 1.5 pts Correct form of equation used, but
$$\Pi_2$$ found correctly
mistake in substituting values (for components of
ÊÊ + 1 pts Normal vectors for $$\Pi_1$$ and
normal vector, coordinates of point)
$$\Pi_2$$ found with minor mistake(s)
ÊÊ + 1 pts Correct general form of equation with
ÊÊ + 0 pts Normal vectors for $$\Pi_1$$ and
