(Modeling) Measuring Rainfall Suppose that vector 𝐑 models the amount of rainfall in inches and

(Modeling) Measuring Rainfall Suppose that vector 𝐑 models...

(Modeling) Measuring Rainfall Suppose that vector $\mathbf{R}$ models the amount of rainfall in inches and the direction it falls, and vector A models the area in square inches and the orientation of the opening of a rain gauge, as illustrated in the figure. The total volume $V$ of water collected in the rain gauge is given by $$V=|\mathbf{R} \cdot \mathbf{A}|$$ This formula calculates the volume of water collected even if the wind is blowing the rain in a slanted direction or the rain gauge is not exactly vertical. Let $\mathbf{R}=\mathbf{i}-2 \mathbf{j}$ and $\mathbf{A}=0.5 \mathbf{i}+\mathbf{j}$ (a) Find $|\mathbf{R}|$ and $|\mathbf{A}|$ to the nearest tenth. Interpret the results. (b) Calculate $V$ to the nearest tenth, and interpret this result.

(Modeling) Measuring Rainfall Suppose that vector $\mathbf{R}$ models the amount of rainfall in inches and the direction it falls, and vector A models the area in square inches and the orientation of the opening of a rain gauge, as illustrated in the figure. The total volume $V$ of water collected in the rain gauge is given by
$$V=|\mathbf{R} \cdot \mathbf{A}|$$
This formula calculates the volume of water collected even if the
wind is blowing the rain in a slanted direction or the rain gauge is not exactly vertical. Let $\mathbf{R}=\mathbf{i}-2 \mathbf{j}$ and $\mathbf{A}=0.5 \mathbf{i}+\mathbf{j}$
(a) Find $|\mathbf{R}|$ and $|\mathbf{A}|$ to the nearest tenth. Interpret the results.
(b) Calculate $V$ to the nearest tenth, and interpret this result.

Key Concepts

Vector Magnitude

Vector magnitude is the measure of a vector's length in space. It is calculated using the Euclidean norm, which involves taking the square root of the sum of the squares of its components. This concept is essential for understanding how measurements such as rainfall intensity and area extend in terms of absolute values without considering direction.

Dot Product

The dot product is an operation that multiplies two vectors and outputs a scalar. It combines both the magnitudes of the vectors and the cosine of the angle between them. This concept is crucial in determining how much one vector extends in the direction of another, such as calculating the effective component of rainfall entering a gauge whose opening might be oriented differently.

Projection

Projection refers to the component of one vector along the direction of another vector. When the dot product is used in this context, it effectively 'projects' one vector onto the other, allowing for the calculation of how much of a force or other quantity influences a given direction. This is particularly useful in physical modeling where the directional effect of an influence, such as wind-blown rain, matters.

Physical Modeling Using Vectors

Physical modeling with vectors enables the representation of directional quantities in a geometric framework. In problems such as calculating the volume of water collected by a rain gauge, vectors allow for the incorporation of both magnitude and direction, ensuring that slanted forces and orientations are accurately accounted for. This approach is foundational in fields like physics and engineering for solving real-world problems.

Transcript

00:01 Now in this problem we are given with this rain gauge and the volume collected in this rain gauge is given by the formula v equals to the magnitude of the dot product between the vector r and a and here are models the amount of rainfall in inches and the direction it falls and the vector a models the area in square inches and the orientation of the opening of the rain goal that you can see in this figure now there are two parts in this question in the a part we need to find the magnitude of our vector and a vector and we are given that our vector we have i minus 2j and the vector a we have 0 .5 i plus j now we can rewrite this vector in this form our vector to be equals to hit the coefficient of i is one so one minus 2 similarly a vector is 0 .5 comma 1 now to find the magnitude of our vector we have square root now to find the magnitude of any vector we have this formula which is square root a square plus b square so here we get 1 square plus minus 2 square which is equals to square root 5 and square root 5 is nearly equal to 2 2 .2 and the magnitude of a vector will be equals to same formula we can use square root 0 .5 square plus 1 square this is equals to square root 0 .25 plus 1 which is equal to square root 1 .25 and if you use a calculator you'll get this value to be nearly 1 .1.

02:03 So this is the magnitude for the r vector and this is the magnitude for the a vector.

02:09 Now next we have to find the volume of water collected in the rain gauge and the volume of water is given by this formula so we have to take the magnitude of the dot product between r and a so first let us find a dot product between r and a vector so r dot a this is equals to 1 comma 2 dot 0 .5 comma 1 so that would be equals to now multiply the x component with the x component and the y component with the y component so we'll get 1 times 0 .5 plus minus 2 times 1 correct so that would be equals to 0 .5 and plus this would be minus 2 times 1 is minus 2 so that is equals to minus 1 .5 now the magnitude of the this value will give the volume...

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