Set the following vectors in the increasing order of their magnitude.
(a) `3hati + 4hatj`, (b) `2hati + 4hatj + 6hatk`
( c) `2hati + 2hatj + 2hatk`,

Options

A. a,b,c
B. c,a,b
C. a,c,b
D. b,c,a

Answer

To solve the problem of arranging the given vectors in increasing order of their magnitude, we will calculate the magnitude of each vector step by step.

### Step 1: Calculate the magnitude of vector (a) `3i + 4j`
The magnitude of a vector \( \vec{A} = ai + bj \) is given by the formula:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
For vector (a):
- \( a = 3 \)
- \( b = 4 \)

Calculating the magnitude:
\[ |\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Calculate the magnitude of vector (b) `2i + 4j + 6k`
For vector (b):
- \( a = 2 \)
- \( b = 4 \)
- \( c = 6 \)

Using the formula for three-dimensional vectors \( \vec{B} = ai + bj + ck \):
\[ |\vec{B}| = \sqrt{a^2 + b^2 + c^2} \]
Calculating the magnitude:
\[ |\vec{b}| = \sqrt{2^2 + 4^2 + 6^2} = \sqrt{4 + 16 + 36} = \sqrt{56} \approx 7.48 \]

### Step 3: Calculate the magnitude of vector (c) `2i + 2j + 2k`
For vector (c):
- \( a = 2 \)
- \( b = 2 \)
- \( c = 2 \)

Calculating the magnitude:
\[ |\vec{c}| = \sqrt{2^2 + 2^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} \approx 3.46 \]

### Step 4: Compare the magnitudes
Now we have the magnitudes:
- \( |\vec{a}| = 5 \)
- \( |\vec{b}| \approx 7.48 \)
- \( |\vec{c}| \approx 3.46 \)

### Step 5: Arrange the vectors in increasing order of their magnitudes
From the calculated magnitudes:
- \( |\vec{c}| \approx 3.46 \) (smallest)
- \( |\vec{a}| = 5 \) (next)
- \( |\vec{b}| \approx 7.48 \) (largest)

Thus, the increasing order of the vectors based on their magnitudes is:
\[ \vec{c}, \vec{a}, \vec{b} \]

### Final Answer
The increasing order of the vectors is \( c, a, b \).

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