Question

Find the vector that has the same direction as 3, 2, −6 but has length 2.

Answer 1

Answer:

The vector is $\vec r = \left(\frac{6}{7},\frac{4}{7},-\frac{12}{7}\right)$.

Step-by-step explanation:

We can determine the equivalent vector ($\vec r$), dimensionless, by means of the following formula:

$\vec r = \frac{\vec u}{\|\vec u\|} \cdot \|\vec r\|$ (1)

Where:

$\vec u$ - Original vector, dimensionless.

$\|\vec u\|$ - Norm of the original vector, dimensionless.

$\|\vec r\|$ - Norm of the new vector, dimensionless.

The norm of the original vector is determined by the following definition:

$\|\vec u\| = \sqrt{\vec u\,\bullet \,\vec u}$ (2)

If we know that $\vec u = (3, 2, -6)$, then the norm of the original vector is:

$\|\vec u\| = \sqrt{(3)\cdot (3)+(2)\cdot (2)+(-6)\cdot (-6)}$

$\|\vec u\| = 7$

If we know that $\|\vec r\| = 2$, then the new vector is:

$\vec r = \frac{2}{7}\cdot (3,2,-6)$

$\vec r = \left(\frac{6}{7},\frac{4}{7},-\frac{12}{7}\right)$

The vector is $\vec r = \left(\frac{6}{7},\frac{4}{7},-\frac{12}{7}\right)$.