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

Solution:

Given, the vector is B(-6, 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 -6u, 4u, 2u.

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

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

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

= √56 u

Given, the length is 6

So, √56 u = 6

u = 6/√56

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

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

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

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

Summary:

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