Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To solve the problem given vectors \( \mathbf{u} = \langle 5, -7 \rangle \) and \( \mathbf{v} = \langle -11, 3 \rangle \), we need to find \( 2\mathbf{v} - 6\mathbf{u} \) and then determine the magnitude (or norm) of that resulting vector. Here is a detailed step-by-step solution:
1. Compute \( 2\mathbf{v} \):
- Multiply each component of vector \(\mathbf{v}\) by 2.
- [tex]\[ 2\mathbf{v} = 2 \langle -11, 3 \rangle = \langle 2 \cdot -11, 2 \cdot 3 \rangle = \langle -22, 6 \rangle \][/tex]
2. Compute \( 6\mathbf{u} \):
- Multiply each component of vector \(\mathbf{u}\) by 6.
- [tex]\[ 6\mathbf{u} = 6 \langle 5, -7 \rangle = \langle 6 \cdot 5, 6 \cdot -7 \rangle = \langle 30, -42 \rangle \][/tex]
3. Compute \( 2\mathbf{v} - 6\mathbf{u} \):
- Subtract each corresponding component of \(6\mathbf{u}\) from \(2\mathbf{v}\).
- [tex]\[ 2\mathbf{v} - 6\mathbf{u} = \langle -22, 6 \rangle - \langle 30, -42 \rangle = \langle -22 - 30, 6 - (-42) \rangle = \langle -52, 48 \rangle \][/tex]
4. Calculate the magnitude (norm) of the resulting vector \( \langle -52, 48 \rangle \):
- The magnitude \( \|\mathbf{a}\| \) of vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is given by \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \).
- [tex]\[ \| 2\mathbf{v} - 6\mathbf{u} \| = \| \langle -52, 48 \rangle \| = \sqrt{(-52)^2 + 48^2} = \sqrt{2704 + 2304} = \sqrt{5008} \approx 70.77 \][/tex]
Hence, the correct answers to fill in the blanks are:
- \( 2\mathbf{v} - 6\mathbf{u} = \langle -52, 48 \rangle \)
- \( \| 2\mathbf{v} - 6\mathbf{u} \| \approx 70.77 \)
Summarizing the filled blanks:
If vector \( u = \langle 5,-7 \rangle \) and \( v = \langle -11, 3 \rangle \):
- \( 2v - 6u = \langle -52, 48 \rangle \)
- [tex]\( \|2v - 6u\| \approx 70.77 \)[/tex]
1. Compute \( 2\mathbf{v} \):
- Multiply each component of vector \(\mathbf{v}\) by 2.
- [tex]\[ 2\mathbf{v} = 2 \langle -11, 3 \rangle = \langle 2 \cdot -11, 2 \cdot 3 \rangle = \langle -22, 6 \rangle \][/tex]
2. Compute \( 6\mathbf{u} \):
- Multiply each component of vector \(\mathbf{u}\) by 6.
- [tex]\[ 6\mathbf{u} = 6 \langle 5, -7 \rangle = \langle 6 \cdot 5, 6 \cdot -7 \rangle = \langle 30, -42 \rangle \][/tex]
3. Compute \( 2\mathbf{v} - 6\mathbf{u} \):
- Subtract each corresponding component of \(6\mathbf{u}\) from \(2\mathbf{v}\).
- [tex]\[ 2\mathbf{v} - 6\mathbf{u} = \langle -22, 6 \rangle - \langle 30, -42 \rangle = \langle -22 - 30, 6 - (-42) \rangle = \langle -52, 48 \rangle \][/tex]
4. Calculate the magnitude (norm) of the resulting vector \( \langle -52, 48 \rangle \):
- The magnitude \( \|\mathbf{a}\| \) of vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) is given by \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \).
- [tex]\[ \| 2\mathbf{v} - 6\mathbf{u} \| = \| \langle -52, 48 \rangle \| = \sqrt{(-52)^2 + 48^2} = \sqrt{2704 + 2304} = \sqrt{5008} \approx 70.77 \][/tex]
Hence, the correct answers to fill in the blanks are:
- \( 2\mathbf{v} - 6\mathbf{u} = \langle -52, 48 \rangle \)
- \( \| 2\mathbf{v} - 6\mathbf{u} \| \approx 70.77 \)
Summarizing the filled blanks:
If vector \( u = \langle 5,-7 \rangle \) and \( v = \langle -11, 3 \rangle \):
- \( 2v - 6u = \langle -52, 48 \rangle \)
- [tex]\( \|2v - 6u\| \approx 70.77 \)[/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.