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Chapter 6: Problem 26

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=8 \mathbf{i}-4 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-12 \mathbf{j}$$

Short Answer

Expert verified
Yes, the vectors \( v \) and \( w \) are orthogonal because their dot product is zero.

Step by step solution

01

Write down vectors

First, express the vectors \( v \) and \( w \) in component form. \( v = 8i - 4j \) can be written as \( v = \[8, -4\] \) and \( w = -6i -12j \) can be written as \( w = \[-6, -12\] \).
02

Compute the dot product.

Second, calculate the dot product of \( v \) and \( w \). The dot product of two vectors \[a, b\] and \[c, d\] is computed as \((a*c) + (b*d)\). Applying this to our vectors \( v \) and \( w \), we get \((8*-6) + (-4*-12) = -48 + 48\).
03

Determine orthogonality.

Finally, determine if \( v \) and \( w \) are orthogonal. Since the dot product of \( v \) and \( w \) is 0, we can conclude that vectors \( v \) and \( w \) are orthogonal because two vectors are orthogonal if their dot product is zero.

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