Questions: Suppose that a boat weighs 700 pounds and is on a ramp inclined at 30 degrees. Represent the force due to gravity, F, using F=-700 j as shown in the figure to the right.
a. Write a unit vector along the ramp in the upward direction.
b. Find the vector projection of F onto the unit vector from part a.
c. What is the magnitude of the vector projection in part b? What does this represent?
a. The unit vector along the ramp in the upward direction is u=
(Simplify your answer. Type your answer in terms of i and j. Use integers or fractions for any numbers in the expression. Rationalize all denominators. Type an exact answer, using radicals as needed.)
Transcript text: Suppose that a boat weighs 700 pounds and is on 'a ramp inclined at $30^{\circ}$. Represent the force due to gravity, $F$, using $F=-700 \mathrm{j}$ as shown in the figure to the right.
a. Write a unit vector along the ramp in the upward direction.
b. Find the vector projection of $F$ onto the unit vector from part
a.
c. What is the magnitude of the vector projection in part b? What does this represent?
a. The unit vector along the ramp in the upward direction is $\mathbf{u}=$ $\square$
(Simplify your answer. Type your answer in terms of i and j . Use integers or fractions for any numbers in the expression. Rationalize all denominators. Type an exact answer, using radicals as needed.)
Solution
Solution Steps
Step 1: Write a unit vector along the ramp in the upward direction
The ramp is inclined at 30 degrees. The unit vector along the ramp in the upward direction can be represented as:
u=cos(30∘)i+sin(30∘)j
Using the trigonometric values:
cos(30∘)=23sin(30∘)=21
Thus, the unit vector is:
u=23i+21j
Step 2: Find the vector projection of F onto the unit vector from part a
The force due to gravity is given by:
F=−700j
The vector projection of F onto u is given by:
projuF=(u⋅uF⋅u)u
First, calculate the dot product F⋅u:
F⋅u=(−700j)⋅(23i+21j)=−700⋅21=−350
Next, calculate the dot product u⋅u:
u⋅u=(23i+21j)⋅(23i+21j)=(23)2+(21)2=43+41=1
Thus, the vector projection is:
projuF=(−350)u=−350(23i+21j)=−1753i−175j
Step 3: What is the magnitude of the vector projection in part b? What does this represent?
The magnitude of the vector projection is given by:
∣projuF∣=(−1753)2+(−175)2
Calculate the squares:
(−1753)2=1752⋅3=91875(−175)2=30625
Add the squares:
91875+30625=122500
Take the square root:
122500=350
The magnitude of the vector projection is 350 pounds. This represents the component of the gravitational force acting along the direction of the ramp.
Final Answer
The unit vector along the ramp in the upward direction is:
u=23i+21j
The vector projection of F onto u is:
projuF=−1753i−175j
The magnitude of the vector projection is 350 pounds, representing the component of the gravitational force acting along the direction of the ramp.