Velocity can be written v = | v | v | v | , where | v | is the speed and v | v | is the unit...

Question:

Velocity can be written {eq}v = |v| \frac{v}{|v|} {/eq}, where {eq}|v| {/eq} is the speed and {eq}\frac{v}{|v|} {/eq} is the unit vector in the direction of motion. For example, a ship at 22 knots traveling NNW {eq}v = 22(-0.1i + 0.995j) {/eq}. Compute the magnitude and the write the vector in the form: {eq}v = |v| \frac{v}{|v|} {/eq} for the following:

a. {eq}v = 4i + 2k {/eq}

b. {eq}v = -8j + 6k {/eq}

Expressing Vectors in different form:

This problem involves expressing the given vectors in the defined form. Since we know the vectors have magnitude and direction, this method helps us write them separately. To find the magnitude we simply take the dot product of the vector with itself and take a root of it. Then, to find the direction, we find the unit vector along the given vector, which is done by dividing the given vector by its magnitude.

Given, the velocity can be written in the form of -

{eq}\displaystyle v = |v| \frac{ v}{|v|} {/eq}

We need to write the given vectors in the above form.

(a) {eq}\displaystyle v = 4 i + 2 k {/eq}

The magnitude of this vector is {eq}\displaystyle |v| = \sqrt { 4^2 + 2^2 } = \sqrt {20} = 2 \sqrt 5 {/eq}

The direction of the vector is -

{eq}\displaystyle \frac{v}{|v|} = \frac{ 4i + 2k}{2 \sqrt 5} = 0.894 i + 0.447 k {/eq}

Thus, the given vector is -

{eq}\displaystyle v = 2\sqrt 5 ( 0.894 i + 0.447 k ) {/eq}

(b) {eq}\displaystyle v = -8 j + 6 k {/eq}

The magnitude of this vector is {eq}\displaystyle |v| = \sqrt { 8^2 + 6^2 } = \sqrt {100} =10 {/eq}

The direction of the vector is -

{eq}\displaystyle \frac{v}{|v|} = \frac{ -8j + 6k}{10} = -0.8 j + 0.6 k {/eq}

Thus, the given vector is -

{eq}\displaystyle v = 10 ( -0.8j +0.6k ) {/eq}