Question11. Solve the followings; a) Given that $\mathrm{P}(-3,4)$ . Calculate the unit vector in the direction of $\overrightarrow{\mathrm{OP}}$ Ans: $\frac{-3 i+4 j}{5}$ b) Given that vectors $\overrightarrow{O P}=-1 i-5 j-11 k$ and $\overrightarrow{O R}=-4 i-2 j+3 k$ .
Express the vectors of $\overrightarrow{P R}$ . Ans: $-3 i+3 j+14 k$

Studdy Solution

STEP 1

1. We are dealing with vector operations in a 2D and 3D space.
2. For part (a), the point $\mathrm{P}(-3,4)$ is considered as a vector from the origin $\mathrm{O}(0,0)$ .
3. For part (b), we are given vectors $\overrightarrow{O P}$ and $\overrightarrow{O R}$ and need to find $\overrightarrow{P R}$ .

STEP 2

1. Calculate the unit vector in the direction of $\overrightarrow{OP}$ .
2. Express the vector $\overrightarrow{PR}$ using given vectors $\overrightarrow{OP}$ and $\overrightarrow{OR}$ .

STEP 3

To find the unit vector in the direction of $\overrightarrow{OP}$ , first determine the magnitude of $\overrightarrow{OP}$ . The vector $\overrightarrow{OP}$ is given by the coordinates of point P, $(-3, 4)$ .
Calculate the magnitude of $\overrightarrow{OP}$ :
$\|\overrightarrow{OP}\| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

STEP 4

Now, divide each component of $\overrightarrow{OP}$ by its magnitude to get the unit vector:
$\text{Unit vector} = \frac{\overrightarrow{OP}}{\|\overrightarrow{OP}\|} = \frac{-3i + 4j}{5}$

STEP 5

To find $\overrightarrow{PR}$ , use the vector subtraction formula:
$\overrightarrow{PR} = \overrightarrow{OR} - \overrightarrow{OP}$
Given: $\overrightarrow{OP} = -1i - 5j - 11k$ $\overrightarrow{OR} = -4i - 2j + 3k$

STEP 6

Subtract the components of $\overrightarrow{OP}$ from $\overrightarrow{OR}$ :
$\overrightarrow{PR} = (-4i - 2j + 3k) - (-1i - 5j - 11k)$
$\overrightarrow{PR} = (-4 + 1)i + (-2 + 5)j + (3 + 11)k$
$\overrightarrow{PR} = -3i + 3j + 14k$
The solutions are: a) $\frac{-3i + 4j}{5}$ b) $-3i + 3j + 14k$

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