Volume of a Fish Tank A fish tank in an avant-garde...

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is $300 \mathrm{cm}$ long and $120 \mathrm{cm}$ wide. The front and back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure $120 \mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.) (a) Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be the three vectors shown in the figure. Find $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ [Hint: Recall that $\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta \text { and }|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta .]$ (b) What is the capacity of the tank in liters? [Note: $\left.1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right]$

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is $300 \mathrm{cm}$ long and $120 \mathrm{cm}$ wide. The front and
back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure $120 \mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.)
(a) Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be the three vectors shown in the figure. Find $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ [Hint: Recall that $\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta \text { and }|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta .]$
(b) What is the capacity of the tank in liters? [Note: $\left.1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right]$

Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. This concept is essential in projecting one vector onto another and is widely used in many areas, including physics and geometry.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in another vector which is perpendicular to the plane of the original vectors. Its magnitude is equal to the area of the parallelogram spanned by the initial vectors and is the product of their magnitudes and the sine of the angle between them. This operation is fundamental in determining orthogonal directions and areas in vector calculus.

Scalar Triple Product

The scalar triple product is a combination of both the dot product and the cross product, given by u · (v × w), and it provides the volume of the parallelepiped formed by three vectors. This product takes into account both the magnitudes of the vectors and the mutual angles between them, resulting in a scalar quantity that signifies the signed volume of the constructed three-dimensional shape.

Volume Calculation of a Parallelepiped

In geometry, the volume of a parallelepiped can be determined using the scalar triple product of three vectors that represent its edges. This method provides a concise and efficient way to compute volumes in three dimensions and is central to solving problems involving three-dimensional shapes with skewed angles or non-orthogonal sides.

Unit Conversion

Unit conversion involves converting quantities from one unit of measurement to another, such as converting cubic centimeters to liters. This is a fundamental skill in mathematics and science, ensuring that calculations remain consistent and comprehensible across different measurement systems.

Transcript

00:01 In this problem, we are given the shown parallel pipe bed, and we are asked to determine what u.

00:08 Dot v cross w is equal to.

00:13 Now, in order to do that, this, we have to know, at least the length of u and v and w.

00:21 And so to do this, the problem tells us that the rectangular base is 300 centimeters long and 120 centimeters wide.

00:37 That means that this is equal to 300 centimeters.

00:50 And this length along the side is equal to 120 centimeters.

01:04 Therefore, we know that the length of vector w is 300 centimeters, and the length of vector u is 120 centimeters.

01:29 And we are also told that the side faces are 120 centimeters, which we just found is u by 150 centimeters, meaning that this length is 150 centimeters.

01:50 And we know over here that that is length of v.

01:53 And so the length of v is equal to 150 centimeters.

02:02 Now we are given a hint about how to find the cross product of two vectors or the dot product of two vectors.

02:10 And so first we're going to try to find the cross product.

02:13 And we're told to find the cross product of v cross w.

02:20 And to do that, we're going to take the length of v.

02:23 We found that's 150 centimeters times the length of w that's 300 centimeters times the sign of the angle between them and what's the angle between v and w we know that the angle between the vertical and v is 30 degrees and we know that the angle between v and w is the rest of that angle which is going to be 60 degrees because the whole angle between the vertical and vector w is 90 degrees and we know that 30 degrees is already taken up, so it's going to be the sign of 60 degrees.

03:03 And so when we do this math, we're going to get, sorry, this is the length of the cross product, v, cross w.

03:17 And so when we do this math, we get 45 ,000 times the sign of 60 degrees, which is going to be root 3 over 2.

03:39 And if you divide 45 ,000 by 2, you're going to get 22 ,500 root 3...

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