Chapter 9: Q. 44 (page 613)

Create an application different from any found in the text that requires a dot product.

Short Answer

Expert verified

We can prove the law of cosines using dot product.

$a^{2} = b^{2} + c^{2} - 2 b c \cos α b^{2} = a^{2} + c^{2} - 2 a c \cos β c^{2} = a^{2} + b^{2} - 2 a b \cos γ$

Step by step solution

Step 1. An application that requires a dot product.

We can prove the law of cosines using dot product.

$a^{2} = b^{2} + c^{2} - 2 b c \cos α b^{2} = a^{2} + c^{2} - 2 a c \cos β c^{2} = a^{2} + b^{2} - 2 a b \cos γ$

Step 2. Proving the law of cosines using dot product.

Let three vectors represent the sides of the triangle.

Let

$||\vec{a}|| = a ||\vec{b}|| = b ||\vec{c}|| = c$

Let angle is $α$ between the $\vec{b}$ and $\vec{c}$ .

Apply triangle law of vector addition,

$\vec{a} = \vec{b} + \vec{c} \vec{a} \cdot \vec{a} = (\vec{b} + \vec{c}) \cdot (\vec{b} + \vec{c}) a^{2} = \vec{b} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{b} + \vec{c} \cdot \vec{c} a^{2} = b^{2} + c^{2} + 2 \vec{b} \cdot \vec{c} a^{2} = b^{2} + c^{2} + 2 ||\vec{b}|| ||\vec{c}|| \cos α a^{2} = b^{2} + c^{2} + 2 b c \cos α$

Similarly we can prove

$b^{2} = a^{2} + c^{2} - 2 a c \cos β c^{2} = a^{2} + b^{2} - 2 a b \cos γ$

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Create an application different from any found in the text that requires a dot product.

Short Answer

Step by step solution

Step 1. An application that requires a dot product.

Step 2. Proving the law of cosines using dot product.

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