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

Solution:

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

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.

VectorB can be written as -2u, 4u, 2u.

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

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

\(\\=\sqrt{((4+16+4)u^{2})}\\=\sqrt{24u^{2}}\)

= √24 u

Given, the length is 6

So, √24 u = 6

u = 6/√24

The vector A = (\(-2\times \frac{6}{\sqrt{24}},4\times \frac{6}{\sqrt{24}},2\times \frac{6}{\sqrt{24}}\))

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

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

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

Summary:

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