Pregunta
\( 1 \leftarrow \quad \) Compute \( \langle-3,3,-6\rangle \cdot\langle 1,4,-6\rangle \)
Ask by Hill Hilton. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The dot product is 45.
Solución
To compute the dot product of the vectors \(\langle -3, 3, -6 \rangle\) and \(\langle 1, 4, -6 \rangle\), you can use the formula for the dot product of two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\):
\[
\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3
\]
Applying this formula to the given vectors:
\[
\langle -3, 3, -6 \rangle \cdot \langle 1, 4, -6 \rangle = (-3)(1) + (3)(4) + (-6)(-6)
\]
Calculate each term:
\[
-3 \times 1 = -3
\]
\[
3 \times 4 = 12
\]
\[
-6 \times -6 = 36
\]
Now, sum these results:
\[
-3 + 12 + 36 = 45
\]
**Answer:** \(45\)
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To compute the dot product of the vectors \(\langle -3, 3, -6 \rangle\) and \(\langle 1, 4, -6 \rangle\), we apply the formula for the dot product: \[ \langle a_1, a_2, a_3 \rangle \cdot \langle b_1, b_2, b_3 \rangle = a_1 b_1 + a_2 b_2 + a_3 b_3 \] Plugging in the values: \[ (-3) \cdot 1 + 3 \cdot 4 + (-6) \cdot (-6) = -3 + 12 + 36 \] Now we simplify this: \[ -3 + 12 + 36 = 9 + 36 = 45 \] Thus, the dot product \(\langle -3, 3, -6 \rangle \cdot \langle 1, 4, -6 \rangle\) is \( \boxed{45} \). This dot product tells us how much one vector extends in the direction of the other: a positive result here indicates they are pointing roughly in the same direction.
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