Given the vectors below, find the dot product.

u=[−1,−2,6]

v=[7,0,−8]

u⋅v=

To find the dot product of two vectors \( \mathbf{u} = [-1, -2, 6] \) and \( \mathbf{v} = [7, 0, -8] \), we use the formula:

\[
\mathbf{u} \cdot \mathbf{v} = (u_1 \cdot v_1) + (u_2 \cdot v_2) + (u_3 \cdot v_3)
\]

Substituting the given values:

\[
\mathbf{u} \cdot \mathbf{v} = (-1 \cdot 7) + (-2 \cdot 0) + (6 \cdot -8)
\]

Now, calculate each term:

\[
-1 \cdot 7 = -7
\]
\[
-2 \cdot 0 = 0
\]
\[
6 \cdot -8 = -48
\]

Adding them together:

\[
-7 + 0 + (-48) = -55
\]

Thus, the dot product \( \mathbf{u} \cdot \mathbf{v} = -55 \).

Would you like more details on this? Here are some related questions:

1. What is the geometric meaning of the dot product?
2. How can we determine if two vectors are perpendicular using the dot product?
3. Can the dot product be negative? Why?
4. What are some applications of the dot product in physics?
5. How is the dot product related to vector projection?

**Tip:** The dot product of two vectors is a scalar, not a vector, and gives information about their directional alignment.

Given the vectors below, find the dot product.
u=[−1,−2,6]
v=[7,0,−8]
u⋅v=

Find the dot product of the vectors u=[−1,−2,6] and v=[7,0,−8] with a detailed step-by-step solution. This problem covers key concepts in vector algebra and the dot product theorem. Suitable for students in Grades 10-12.

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