Math / Geometry

Question(ii) The direction ratios of a line are $(-6,2,3)$ and its direction cosines

Studdy Solution

STEP 1

1. Direction ratios of a line are given as $(-6, 2, 3)$ .
2. Direction cosines are the cosines of the angles that the line makes with the coordinate axes.
3. Direction cosines can be found by normalizing the direction ratios.

STEP 2

1. Calculate the magnitude of the direction ratios vector.
2. Normalize the direction ratios to obtain the direction cosines.

STEP 3

Calculate the magnitude of the direction ratios vector $(-6, 2, 3)$ .
$\text{Magnitude} = \sqrt{(-6)^2 + 2^2 + 3^2}$

STEP 4

Simplify the expression under the square root.
$\text{Magnitude} = \sqrt{36 + 4 + 9}$

STEP 5

Add the values inside the square root.
$\text{Magnitude} = \sqrt{49}$

STEP 6

Take the square root to find the magnitude.
$\text{Magnitude} = 7$

STEP 7

Normalize the direction ratios by dividing each component by the magnitude to obtain the direction cosines.
$\text{Direction Cosines} = \left( \frac{-6}{7}, \frac{2}{7}, \frac{3}{7} \right)$

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Question(ii) The direction ratios of a line are $(-6,2,3)$ and its direction cosines

Studdy Solution

STEP 1

1. Direction ratios of a line are given as $(-6, 2, 3)$ .
2. Direction cosines are the cosines of the angles that the line makes with the coordinate axes.
3. Direction cosines can be found by normalizing the direction ratios.

STEP 2

1. Calculate the magnitude of the direction ratios vector.
2. Normalize the direction ratios to obtain the direction cosines.

STEP 3

Calculate the magnitude of the direction ratios vector $(-6, 2, 3)$ .
$\text{Magnitude} = \sqrt{(-6)^2 + 2^2 + 3^2}$

STEP 4

Simplify the expression under the square root.
$\text{Magnitude} = \sqrt{36 + 4 + 9}$

STEP 5

Add the values inside the square root.
$\text{Magnitude} = \sqrt{49}$

STEP 6

Take the square root to find the magnitude.
$\text{Magnitude} = 7$

STEP 7

Normalize the direction ratios by dividing each component by the magnitude to obtain the direction cosines.
$\text{Direction Cosines} = \left( \frac{-6}{7}, \frac{2}{7}, \frac{3}{7} \right)$

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Question(ii) The direction ratios of a line are (−6,2,3)(-6,2,3)(−6,2,3) and its direction cosines

Studdy Solution

STEP 1

1. Direction ratios of a line are given as (−6,2,3)(-6, 2, 3)(−6,2,3).2. Direction cosines are the cosines of the angles that the line makes with the coordinate axes.3. Direction cosines can be found by normalizing the direction ratios.

STEP 2

1. Calculate the magnitude of the direction ratios vector.2. Normalize the direction ratios to obtain the direction cosines.

STEP 3

Calculate the magnitude of the direction ratios vector (−6,2,3)(-6, 2, 3)(−6,2,3).Magnitude=(−6)2+22+32 \text{Magnitude} = \sqrt{(-6)^2 + 2^2 + 3^2} Magnitude=(−6)2+22+32​

STEP 4

Simplify the expression under the square root.Magnitude=36+4+9 \text{Magnitude} = \sqrt{36 + 4 + 9} Magnitude=36+4+9​

STEP 5

Add the values inside the square root.Magnitude=49 \text{Magnitude} = \sqrt{49} Magnitude=49​

STEP 6

Take the square root to find the magnitude.Magnitude=7 \text{Magnitude} = 7 Magnitude=7

STEP 7

Normalize the direction ratios by dividing each component by the magnitude to obtain the direction cosines.Direction Cosines=(−67,27,37) \text{Direction Cosines} = \left( \frac{-6}{7}, \frac{2}{7}, \frac{3}{7} \right) Direction Cosines=(7−6​,72​,73​)

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Question(ii) The direction ratios of a line are $(-6,2,3)$ and its direction cosines

1. Direction ratios of a line are given as $(-6, 2, 3)$ .
2. Direction cosines are the cosines of the angles that the line makes with the coordinate axes.
3. Direction cosines can be found by normalizing the direction ratios.

1. Calculate the magnitude of the direction ratios vector.
2. Normalize the direction ratios to obtain the direction cosines.

Calculate the magnitude of the direction ratios vector $(-6, 2, 3)$ .
$\text{Magnitude} = \sqrt{(-6)^2 + 2^2 + 3^2}$

Simplify the expression under the square root.
$\text{Magnitude} = \sqrt{36 + 4 + 9}$

Add the values inside the square root.
$\text{Magnitude} = \sqrt{49}$

Take the square root to find the magnitude.
$\text{Magnitude} = 7$

Normalize the direction ratios by dividing each component by the magnitude to obtain the direction cosines.
$\text{Direction Cosines} = \left( \frac{-6}{7}, \frac{2}{7}, \frac{3}{7} \right)$