please explain how to solve

Question

An airplane needs to head due north, but there is a wind blowing from the southwest at 90 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane? Round to the nearest tenth.

An airplane is heading north at an airspeed of 550 km/hr, but there is a wind blowing from the southwest at 60 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground? Round to the nearest tenth.

A woman leaves home and walks 3 miles west, then 4 miles southwest. How far from home is she, and in what direction must she walk to head directly home? Round to the nearest tenth.

Find the magnitude of the horizontal and vertical components of the vector with magnitude 8 pounds pointed in a direction of 136∘. Round to the nearest hundredth.

A 32 pound box is resting on a ramp that is inclined 8∘. Rounding to the nearest tenth, answer the questions.

Find the magnitude of the normal (perpendicular) component of the force

Find the magnitude of the component of the force that is parallel to the ramp.

Answer 1

To answer these problems, you will need to utilize the different properties of trigonometry of sine, cosine, and tangent, concepts within polar coordinates, and interpreting/applying vector components.

Problem #1 will definitely use a mixture of vector components and trig techniques with sine and cosine.

The phrase SOH CAH TOA will come in handy to know what component of each vector gets sine or cosine. SINE is NOT ALWAYS for Y-DIRECTION and COSINE is NOT ALWAYS for X-DIRECTION! Base it off of the right triangle drawing you construct of your vectors!

Also, you won't know be able to know the magnitude of the resultant speed to the northern destination until you figure out the angle of the plane.

Lastly for problem #1, the inverse cosine function is WAY more reliable in most cases than inverse sine.

If done correctly, you should get an answer of approximately 6.64º West of North (96.64º from +x-axis).

For Problem #2, it will require vector components again and converting from Cartesian coordinates to Polar coordinates.

If you have a vector like the following: A i + B j, then:

> To get the direction, θ = tan^-1 ( B / A )

> To get the magnitude, M = √( A² + B² )

Done correctly, the direction and magnitude will be 4.10º East of North (85.90º from +x-axis) and 593.94 km/hr, respectively.

For Problem #3, it will use the same techniques used in Problem #2.

Done correctly, you'll get a magnitude and direction of 6.48 miles and 205.89º from +x-axis (or 25.89º from -x-axis).

For Problem #4, it too will use the same techniques from Problem #2 and Problem #3. Strangely easier than #2 and #3 though.

Done correctly, you'll get horizontal and vertical magnitudes of 5.75 lbs and 5.56 lbs (note, when you directly compute it, you'll originally get -5.75 and +5.56; however, magnitude does NOT care about negative signs, take the absolute value of any negatives.

For Problem #5, it will combine all prior techniques and an understanding with incline plane concepts.

The slope that the block is resting on is 8º but gravity acts straight down. This will require a rotation of your coordinate axes.

A similar triangle of the inclined plane can be made with the weight due to gravity vector.

The weight will be the hypotenuse of the right triangle, the opposite of the angle with be along the slope, and the adjacent will be normal to the slope (or perpendicular to the slope, normal and perpendicular are synonymous and interchangeable in most cases; I haven't run into a case when it isn't but anyway)

Done correctly, the normal and along slope magnitudes will be 31.69 lbs and 4.45 lbs, respectively.

I hope this helps! Message me in the comments with any questions, comments, or concerns about anything that I explained above!