document.write( "Question 1163755: Let u = [2, −3, 1] and v = [0, 1, 4]. Calculate the vector u − v. Place u and v tail-to-tail to form two sides of a triangle. With regard to this triangle, what does u − v represent? Calculate the number u•u and discuss its relevance to the diagram you have drawn. Do the same for the number (u − v)• (u − v). \n" ); document.write( "
Algebra.Com's Answer #854371 by CPhill(2264)\"\" \"About 
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### 1. Calculation of $u - v$
\n" ); document.write( "To subtract vector $v$ from $u$, we subtract the corresponding components:
\n" ); document.write( "$$u - v = [2 - 0, \ -3 - 1, \ 1 - 4] = [2, \ -4, \ -3]$$\r
\n" ); document.write( "\n" ); document.write( "### 2. Geometric Representation
\n" ); document.write( "When $u$ and $v$ are placed **tail-to-tail**, they form two sides of a triangle.
\n" ); document.write( "* The vector $u - v$ represents the **third side** of that triangle.
\n" ); document.write( "* Specifically, it is the vector that points from the **head of $v$** to the **head of $u$**.\r
\n" ); document.write( "\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "### 3. Analysis of $u \cdot u$
\n" ); document.write( "**Calculation:**
\n" ); document.write( "$$u \cdot u = (2)^2 + (-3)^2 + (1)^2 = 4 + 9 + 1 = 14$$\r
\n" ); document.write( "\n" ); document.write( "**Relevance to the Diagram:**
\n" ); document.write( "The dot product of a vector with itself is equal to the square of its magnitude ($u \cdot u = |u|^2$).
\n" ); document.write( "* In your triangle, $u \cdot u$ represents the **squared length of the side formed by vector $u$**.
\n" ); document.write( "* To find the actual length of that side of the triangle, you would take $\sqrt{14} \approx 3.74$.\r
\n" ); document.write( "\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "### 4. Analysis of $(u - v) \cdot (u - v)$
\n" ); document.write( "**Calculation:**
\n" ); document.write( "Using our result for $u - v = [2, -4, -3]$:
\n" ); document.write( "$$(u - v) \cdot (u - v) = (2)^2 + (-4)^2 + (-3)^2 = 4 + 16 + 9 = 29$$\r
\n" ); document.write( "\n" ); document.write( "**Relevance to the Diagram:**
\n" ); document.write( "Similar to $u \cdot u$, this represents the **squared length of the third side of the triangle** (the side connecting the tips of $u$ and $v$).
\n" ); document.write( "* $|u - v|^2 = 29$.
\n" ); document.write( "* The length of this side is $\sqrt{29} \approx 5.39$.\r
\n" ); document.write( "\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "### Summary Table\r
\n" ); document.write( "\n" ); document.write( "| Expression | Vector Result | Scalar Value (Dot Product) | Geometric Meaning |
\n" ); document.write( "| :--- | :--- | :--- | :--- |
\n" ); document.write( "| **$u$** | $[2, -3, 1]$ | $14$ | Squared length of side 1 |
\n" ); document.write( "| **$v$** | $[0, 1, 4]$ | $17$ | Squared length of side 2 |
\n" ); document.write( "| **$u - v$** | $[2, -4, -3]$ | $29$ | Squared length of side 3 |\r
\n" ); document.write( "\n" ); document.write( "> **Note:** If you apply the Law of Cosines to this triangle ($c^2 = a^2 + b^2 - 2ab \cos \theta$), you'll find that these dot products are exactly the $a^2, b^2,$ and $c^2$ terms!\r
\n" ); document.write( "\n" ); document.write( "Would you like to calculate the angle $\theta$ between $u$ and $v$ using these values? \n" ); document.write( "
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