Vectors and Operations

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Questions and Answers

Given vectors a, b, c, and d in a coordinate plane, describe how to find the resultant vector c - b + 3d geometrically.

First, find the vector -b by reversing the direction of b. Then, scale d by a factor of 3 to obtain 3d. Next, add c, -b, and 3d tip-to-tail to find the resultant vector.

A vector has a magnitude of 16 and a direction of 230. Describe the signs of the x and y components and why they are those signs.

Both the x and y components are negative because 230 lies in the third quadrant, where both cosine (adjacent/x) and sine (opposite/y) values are negative.

Explain why the linear combination form of a vector with magnitude 90 in the direction of (-3, -$\sqrt{3}$) involves rationalizing the denominator.

The linear combination form involves a unit vector in the given direction. Calculating this unit vector requires dividing by the magnitude of (-3, -$\sqrt{3}$), which results in a radical in the denominator that must be rationalized.

If the component form of a vector with magnitude 9 in the direction of (4, -3) is calculated, why is it important to consider the direction vector as a ratio?

The direction vector (4, -3) represents the ratio of the x and y components. To find the actual components, this ratio must be scaled proportionally to match the magnitude of 9. Signup and view all the answers

Explain why the unit vector in the same direction as (8, -15) always has a magnitude of 1.

A unit vector is defined as a vector with a magnitude of 1. It's obtained by dividing a given vector by its original magnitude, effectively normalizing the vector to unit length. Signup and view all the answers

Describe how the dot product is used to calculate the angle between two vectors (4, -1) and (-1, -4).

The dot product of (4, -1) and (-1, -4) is calculated. Then the magnitudes of each vector are calculated. These values are then used in the formula: $cos \theta = \frac{a \cdot b}{||a|| \cdot ||b||}$ to find the angle $ \theta $. Signup and view all the answers

A plane's true speed and direction are affected by wind. If a plane is flying N80E but there is wind blowing from S80E, how does this affect the plane? Be specific.

The wind blowing from S80E will decrease the plane's eastward component and slightly decrease its northward component. The true speed will likely be lower than the airspeed, and the true direction will be slightly more north than the original direction. Signup and view all the answers

When calculating the wind's speed and direction affecting Genevieve's hang gliding, why is vector subtraction used differently than scalar subtraction?

Vector subtraction accounts for both the magnitude and direction of the velocities. Scalar subtraction only considers the magnitudes, ignoring the directional components, which are crucial for accurate vector addition. Signup and view all the answers

Explain how resolving forces into components simplifies finding the resultant force when three forces act on an object at different angles.

Resolving each force into x and y components allows you to add all the x-components together and all the y-components together. This simplifies the problem into adding two vectors (the sum of the x-components and the sum of the y-components), which then result in the resultant vector. Signup and view all the answers

If an Ewok is suspended by two cables, how does increasing the angle between the cables affect the tension in each cable, assuming the Ewok's weight remains constant?

Increasing the angle between the cables increases the tension in each cable. As the angle increases, the vertical components of the tension in each cable must still sum to the Ewok's weight, requiring a larger overall tension in each cable. Signup and view all the answers

Describe how the angle between the forces is important in determining is Cooper and Reid can move the stump.

The total force applied depends on the angle. If they pull in the same direction (0 angle), the forces add directly. If they pull in opposite directions (180 angle), the forces subtract. Any angle between 0 and 180 will result in a total force between the sum and the difference of the individual forces. Signup and view all the answers

In the speedboat problem, is it expected that the wind and current are going in the same direction? Why or why not?

It is unlikely as they are separate systems. The problem requires applying vector addition to determine how both currents affect the boat. Signup and view all the answers

Explain how the Law of Cosines is applicable when determining the distance and bearing for the boat's return trip.

The Law of Cosines applies because the boat's outbound and second leg of the journey form two sides of a triangle, and the angle between these legs can be determined. This allows for calculating the length of the third side (the direct return trip) using the Law of Cosines. Signup and view all the answers

If a squad travels N 60W and then S 40W, explain why you can't simply add the distances to find the total distance from the base.

The distances can't be added because the directions are different. These directions must be treated as vectors, and the distances covered in each direction must be considered as vector components to find the resultant displacement. Signup and view all the answers

How does bearing relate to direction in aviation and why is it important to state the reference from where the bearing is measured?

Bearing is the angle measured clockwise from North or South, representing the direction of travel. It is important to state the reference (North or South) because the same angle measured from North and South will give completely opposite directions. Signup and view all the answers

Explain why the Law of Cosines is important to finding the angle made by the two slanted sections of the roof.

By recognizing that the roof forms a triangle with the base of the shed, and all three sides of the triangles are known, we can use Law of Cosines to find the angle. Signup and view all the answers

Describe a scenario where finding the area of a quadrilateral by dividing it into triangles is more efficient than using a single formula.

When the quadrilateral is irregular and doesn't fit a standard formula, dividing it into triangles allows us to use trigonometric formulas specifically for triangles (like $1/2 \cdot a \cdot b \cdot sin(C)$) to find the area of each triangle and then sum the areas. Signup and view all the answers

Explain the difference between vector projection and scalar projection (component) and, to solidify your explanation, provide different real-world examples of these two projections.

Vector projection results in a vector, representing the vector component of one vector along the direction of another. Scalar projection results in a scalar, representing the magnitude of that component. Vector projection ex: force vector along specific direction or a displacement vector along a specific axis. Scalar projection ex: work done by a force along a displacement. Signup and view all the answers

Describe how you would find the area of a triangle using Heron's formula and when this method would be most appropriate.

Heron's formula calculates the area of a triangle when all three side lengths are known. First, calculate the semi-perimeter (s) as half the sum of the sides. Then, the area is $ \sqrt{s(s-a)(s-b)(s-c)}$, where a, b, and c are the side lengths. This is useful when angles are not known, only side lengths are. Signup and view all the answers

Explain how the Law of Sines is used in navigation to determine unknown distances or angles, and why it's necessary to have at least one side and its opposite angle known.

The Law of Sines relates the sides of a triangle to the sines of their opposite angles ($\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$). In navigation, if you know one side and its opposite angle, along with another angle or side, you can solve for unknown distances or angles, which is useful to finding a position. Signup and view all the answers

Flashcards

What is vector magnitude?

A vector's magnitude is its length. Represented as ||v||, it is a scalar value.

What is a unit vector?

A vector with a magnitude (length) of 1.

What is linear combination of vectors?

Representing a vector as a sum of scalar multiples of basis vectors (i, j, k).

What is bearing?

The direction to or from which something is headed, expressed as an angle relative to North or South.

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What is true course/resultant?

The resultant vector representing the actual path and speed of a boat or plane.

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What is actual speed?

The actual speed of a boat or plane through the air or water, disregarding any external factors like wind or current.

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Study Notes