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12 grade maths others

Find the angle between the lines with direction cosines 1/2, -1/3, 1/4 and 1/3, 1, 2/3.

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10 Months agoGrade
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ApprovedApproved Tutor Answer10 Months ago

To find the angle between two lines given their direction cosines, we can use the formula involving the dot product of the direction cosines. The direction cosines for the first line are \( l_1 = \frac{1}{2} \), \( m_1 = -\frac{1}{3} \), and \( n_1 = \frac{1}{4} \). For the second line, the direction cosines are \( l_2 = \frac{1}{3} \), \( m_2 = 1 \), and \( n_2 = \frac{2}{3} \).

Step-by-Step Calculation

1. **Calculate the dot product** of the direction cosines:

The dot product formula is:

Dot Product = l_1 * l_2 + m_1 * m_2 + n_1 * n_2

Substituting the values:

Dot Product = \left(\frac{1}{2} \cdot \frac{1}{3}\right) + \left(-\frac{1}{3} \cdot 1\right) + \left(\frac{1}{4} \cdot \frac{2}{3}\right)

Calculating each term:

  • First term: \( \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \)
  • Second term: \( -\frac{1}{3} \cdot 1 = -\frac{1}{3} \)
  • Third term: \( \frac{1}{4} \cdot \frac{2}{3} = \frac{1}{6} \)

Now, combine these results:

Dot Product = \frac{1}{6} - \frac{1}{3} + \frac{1}{6} = \frac{1}{6} - \frac{2}{6} + \frac{1}{6} = 0

Finding the Angle

2. **Use the dot product to find the cosine of the angle**:

The formula for the cosine of the angle \( \theta \) between the two lines is:

cos(θ) = (Dot Product) / (Magnitude of Line 1 * Magnitude of Line 2)

3. **Calculate the magnitudes** of the direction cosines:

Magnitude of Line 1:

Magnitude = √(l_1² + m_1² + n_1²) = √\left(\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2\right)

Calculating:

  • First term: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
  • Second term: \( \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \)
  • Third term: \( \left(\frac{1}{4}\right)^2 = \frac{1}{16} \)

Finding a common denominator (144):

Magnitude = √\left(\frac{36}{144} + \frac{16}{144} + \frac{9}{144}\right) = √\left(\frac{61}{144}\right) = \frac{√61}{12}

Magnitude of Line 2:

Magnitude = √(l_2² + m_2² + n_2²) = √\left(\left(\frac{1}{3}\right)^2 + (1)^2 + \left(\frac{2}{3}\right)^2\right)

Calculating:

  • First term: \( \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
  • Second term: \( 1^2 = 1 = \frac{9}{9} \)
  • Third term: \( \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)

Finding a common denominator (9):

Magnitude = √\left(\frac{1}{9} + \frac{9}{9} + \frac{4}{9}\right) = √\left(\frac{14}{9}\right) = \frac{√14}{3}

Final Calculation

4. **Substituting into the cosine formula**:

cos(θ) = 0 / \left(\frac{√61}{12} \cdot \frac{√14}{3}\right) = 0

Since the cosine of the angle is 0, this means:

θ = 90°

The angle between the two lines is 90 degrees, indicating they are perpendicular.


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Step 1: Building Number Understanding

Before children start solving sums, they must first understand numbers clearly.

Young learners should become familiar with:

  • Counting numbers

  • Recognising numbers

  • Understanding the value of numbers

Teachers often use simple activities such as counting objects, toys, or fruits to help children understand numbers.

For example, parents can ask questions like:

  • How many apples are on the table?

  • Can you count the pencils in your bag?

These small activities help children connect numbers with real objects.

Step 2: Learning Addition (Basic Sums)

Addition is usually the first mathematical operation children learn.

Addition means combining numbers together.

For example:

2 + 3 = 5

Teachers often teach addition using objects.

Example:

If a child has 2 pencils and receives 3 more pencils, the total becomes 5 pencils.

Using objects makes the concept easier to understand.

Simple addition practice

Parents can practice with children using:

  • fingers

  • small toys

  • beads

  • fruits

Once children understand the concept, they can move on to written sums.

Step 3: Understanding Subtraction

Subtraction means taking away from a number.

Example:

5 − 2 = 3

Teachers often explain subtraction using real situations.

Example:

If a child has 5 chocolates and gives 2 chocolates to a friend, how many chocolates remain?

This approach helps children understand subtraction more clearly.

Parents can encourage children to think about subtraction in everyday situations.

Step 4: Learning Multiplication

Multiplication is often introduced after children become comfortable with addition.

Multiplication means repeated addition.

Example:

3 × 4 means adding 4 three times.

4 + 4 + 4 = 12

Teachers often explain multiplication using groups.

Example:

If there are 3 groups of 4 apples, the total number of apples is 12.

Using pictures or objects helps children visualise multiplication easily.

Step 5: Multiplication Tables (Learning Vaaipaadu)

In Tamil education systems, multiplication tables are often called “vaaipaadu”.

Memorising tables helps children solve multiplication problems faster.

Common tables include:

2 × 1 = 2
2 × 2 = 4
2 × 3 = 6

Tables usually begin from 2 and go up to 10 or 12.

Teachers often encourage children to practice tables through:

  • repetition

  • rhythm or songs

  • daily practice

Parents can make table learning interesting by asking questions during daily activities.

For example:

If one box contains 5 chocolates, how many chocolates are there in 3 boxes?

These small questions strengthen table knowledge.

Step 6: Understanding Division

Division is the opposite of multiplication. It means sharing or splitting numbers equally.

Example:

12 ÷ 3 = 4

Teachers explain division using sharing examples.

Example:

If 12 sweets are shared equally among 3 children, each child receives 4 sweets.

This helps children understand the concept of division clearly.

Making Basic Maths Skills Practice Fun at Home

Mathematics should not feel like a stressful subject for children. Parents can help children enjoy maths through simple activities.

Some useful methods include:

Asking children to count objects around the house

Practicing tables during daily routines

Solving small sums during shopping or cooking

Playing number-based games

When maths becomes part of daily life, children learn faster.

Encouraging Confidence in Maths

Some children feel nervous about mathematics because they fear making mistakes.

Teachers usually encourage students to see mistakes as part of learning.

Parents can support children by:

  • appreciating their effort

  • encouraging them to try again

  • avoiding comparison with other students

Confidence grows when children feel supported.

Balancing Practice and Understanding

Memorising formulas or tables alone is not enough. Children should also understand the logic behind the calculations.

For example, a child who understands multiplication as repeated addition will find division easier to learn.

Teachers often recommend a balanced approach:

  • understanding concepts

     

  • regular practice

     

  • problem-solving exercises

This combination helps children build strong mathematical skills.

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Grade 1212 grade maths others

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