Find a vector that has the same direction as -2, 6, 4 but has length 6.

Solution:

Given, the vector is B(-2, 6, 4)

We have to find the vector that has the same direction as the given vector but has length 6.

Let A be the vector in the same direction as B but with length 6.

Vector B can be written as <-2u, 6u, 4u>

The length of vector B = \(\sqrt{(-2u)^{2}+(6u)^{2}+(4u)^{2}}\)

\(\sqrt{(-2u)^{2}+(6u)^{2}+(4u)^{2}}=\sqrt{(4u^{2}+36u^{2}+16u^{2})}\)

\(\\=\sqrt{((4+36+16)u^{2})}\\=\sqrt{56u^{2}}\)

= √56 u

Given, the length is 6

So, √56 u = 6

u = 6/√56

The vector A = (\(-2\times \frac{6}{\sqrt{56}},6\times \frac{6}{\sqrt{56}},4\times \frac{6}{\sqrt{56}}\))

Vector A = \(\frac{-12}{\sqrt{56}},\frac{36}{\sqrt{56}},\frac{24}{\sqrt{56}}\)

Therefore, vector A = \(\frac{-12}{\sqrt{56}},\frac{36}{\sqrt{56}},\frac{24}{\sqrt{56}}\).

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Find a vector that has the same direction as -2, 6, 4 but has length 6.

Summary:

A vector that has the same direction as -2, 6, 4 but has length 6 is (\(\frac{-12}{\sqrt{56}},\frac{36}{\sqrt{56}},\frac{24}{\sqrt{56}}\)).