To find all values [tex]\(x\)[/tex] such that the magnitude (or norm) of the vector [tex]\(v + w\)[/tex] equals 7, follow these steps:
1. Express the vectors in components:
[tex]\[
v = 4i - 4j
\][/tex]
[tex]\[
w = xi + 9j
\][/tex]
2. Add the vectors:
[tex]\[
v + w = (4i - 4j) + (xi + 9j) = (4 + x)i + (-4 + 9)j = (4 + x)i + 5j
\][/tex]
3. Find the magnitude of [tex]\(v + w\)[/tex]:
The magnitude of a vector [tex]\((a, b)\)[/tex] is given by [tex]\(\sqrt{a^2 + b^2}\)[/tex]:
[tex]\[
\| v + w \| = \sqrt{(4 + x)^2 + 5^2} = \sqrt{(4 + x)^2 + 25}
\][/tex]
4. Set up the equation for the magnitude:
We need the magnitude to be equal to 7:
[tex]\[
\sqrt{(4 + x)^2 + 25} = 7
\][/tex]
5. Square both sides of the equation to eliminate the square root:
[tex]\[
(4 + x)^2 + 25 = 49
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
(4 + x)^2 + 25 = 49
\][/tex]
Subtract 25 from both sides:
[tex]\[
(4 + x)^2 = 24
\][/tex]
Take the square root of both sides:
[tex]\[
4 + x = \pm \sqrt{24}
\][/tex]
Simplify [tex]\(\sqrt{24}\)[/tex]:
[tex]\[
\sqrt{24} = 2\sqrt{6}
\][/tex]
Therefore, we have:
[tex]\[
4 + x = 2\sqrt{6} \quad \text{or} \quad 4 + x = -2\sqrt{6}
\][/tex]
7. Solve for [tex]\(x\)[/tex] in each case:
[tex]\[
4 + x = 2\sqrt{6}
\][/tex]
[tex]\[
x = 2\sqrt{6} - 4
\][/tex]
and
[tex]\[
4 + x = -2\sqrt{6}
\][/tex]
[tex]\[
x = -2\sqrt{6} - 4
\][/tex]
So, the values of [tex]\(x\)[/tex] that satisfy the condition [tex]\(\| v + w \| = 7\)[/tex] are:
[tex]\[
x = 2\sqrt{6} - 4 \quad \text{or} \quad x = -2\sqrt{6} - 4
\][/tex]
