Respuesta :
Answer:
As per the given statement:
Let w ∈[tex]R^2[/tex] i.e w = (a, b)
then,
by perpendicular to v means;
[tex]w \cdot v=0[/tex]
[tex](a, b) \cdot (7, 8)=0[/tex]
[tex]7a + 8b = 0[/tex]
or
[tex]7a= -8b[/tex]
Divide both sides by 7 we get;
[tex]a = -\frac{8}{7}b[/tex] ......[1]
so, [tex]w = (-\frac{8}{7}b, b)[/tex] for some values of b.
Using the fact: ||w|| = 7 we get;
[tex]||w|| = \sqrt{(-\frac{8}{7}b)^2+b^2 }[/tex]
substitute the given values, we have;
[tex]7= \sqrt{(-\frac{8}{7}b)^2+b^2}[/tex]
Squaring both sides we have;
[tex]7^2 =\frac{64}{49}b^2+b^2[/tex]
Simplify:
[tex]49 = \frac{113}{49}b^2[/tex]
Or
[tex]b^2 = 49 \times \frac{49}{113}[/tex]
or
[tex]b = \pm \sqrt{49 \times \frac{49}{113}}=7 \cdot \frac{7}{\sqrt{113}} =\pm\frac{49}{\sqrt{113}}[/tex]
Substitute the value of b in [1] we get;
[tex]a =- \frac{8}{7}b =-\frac{8}{7} \cdot \frac{49}{\sqrt{113}} = - \frac{56}{\sqrt{113}}[/tex]
therefore, two such answers of w are;
[tex]w = \pm (-\frac{56}{\sqrt{113}}, \frac{49}{\sqrt{113}})[/tex]
By using the fact that w is perpendicular to v, and that we know its magnitude, we will see that w = (-5.27, 4.61 ).
How to find the vector w?
We know that the vector w is something like:
w = (a, b)
And this is such that:
|w| = 7, and it is perpendicular to v = (7, 8).
Remember that two vectors are perpendicular if the dot product between them is 0, so we must have:
w·v = 0 = (a, b)·(7, 8) = a*7 + b*8
Then we will get:
a*7 = -b*8
a = b*(-8/7)
Ok, now we need to use the magnitude equation:
|w| = 7 = √(a^2 + b^2)
Replacing a, we get:
7 = √((b*(-8/7))^2 + b^2) = b*√( 64/49 + 1)
7 = b*1.52
7/1.52 = b = 4.61
Then the value of a is:
a = b*(-8/7) = 4.61*(-8/7) = -5.27
Concluding, the vector w is:
w = (-5.27, 4.61 )
If you want to learn more about vectors, you can read:
https://brainly.com/question/3184914
