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

Solution:

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

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, 6u.

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

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

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

= √56 u

Given, the length is 6

So, √56 u = 6

u = 6/√56

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

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

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

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

Summary:

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