Question

find the velocity having magnitude 15 and parallel to 3icap+2jcap-6kcap​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a vector
• $\sf{\vec{r} = 3 \hat{i} + 2 \hat{j} - 6 \hat{k}}$

To Find:

• We have to find a velocity vector having magnitude 15 and parallel to given vector

Concept Used:

A Quantity having both magnitude as well as direction is known as a vector quantity

$\boxed{\sf{\red{\vec{A} = |A| \times \hat{A}}}}$

Where

• $\sf{|A|}$ = Magnitude of A vector
• $\sf{\hat{A}}$ = Unit vector of A which the represent the direction of vector

Assuming a general vector A which is given below :

$\sf{\vec{A} = a \hat{i} + b \hat{j} + c \hat{k}}$

Then the magnitude of given A vector is calculated by the formula

$\boxed{\sf{|A| = \sqrt{(a)^2 + (b)^2 + (c)^2}}}$

Solution:

We have been given a vector :

$\sf{\vec{r} = 3 \hat{i} + 2 \hat{j} - 6 \hat{k}}$

$\underline{\large\mathfrak\purple{Magnitude \: of \: Vector \: r \: is}}$

$\implies \sf{| \: r \: | = \sqrt{(3)^2 + (2)^2 + (-6)^2}}$

$\implies \sf{| \: r \: | = \sqrt{9 + 4 + 36}}$

$\implies \sf{| \: r \: | = \sqrt{49}}$

$\implies \boxed{\sf{| \: r \: | = 7}}$

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Finding the unit vector in direction of given r vector

$\implies \boxed{\sf{\hat{r} = \dfrac{\vec{r}}{|r|}}}$

$\implies \sf{\hat{r} = \left ( \dfrac{3 \hat{i} + 2 \hat{j} - 6 \hat{k}}{7} \right ) }$

Unit vector in direction of r vector is

$\boxed{\sf{\hat{r} = \left ( \dfrac{3 \hat{i} + 2 \hat{j} - 6 \hat{k}}{7} \right ) }}$

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$\underline{\large\mathfrak\purple{Finding \: value \: of \: velocity \: vector:}}$

Magnitude = 15

Parallel to given vector i.e.

Unit vector of v = Unit vector of r

$\implies \sf{\vec{v} = |v| \times \hat{v}}$

$\implies \sf{\vec{v} = 15 \times \hat{r}}$

$\implies \sf{\vec{v} = 15 \times \left ( \dfrac{3 \hat{i} + 2 \hat{j} - 6 \hat{k}}{7} \right )}$

$\implies \sf{\vec{v} = \dfrac{15}{7} \times (3 \hat{i} + 2 \hat{j} - 6 \hat{k} ) }$

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$\huge\underline{\sf{\red{A}\orange{n}\green{s}\pink{w}\blue{e}\purple{r}}}$

$\large\boxed{\sf{\red{Velocity = \dfrac{15}{7} \: (3 \hat{i} + 2 \hat{j} - 6 \hat{k} )}}}$

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