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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.4

Question 1.

Find the magnitude of

Solution:

Question 2.

Show that

Solution:

Question 3.

Find the vectors of magnitude \(10 \sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2 \hat{j}+\hat{k}\) and \(\hat{i}+3 \hat{j}+4 \hat{k}\)

Solution:

Question 4.

Find the unit vectors perpendicular to each of the vectors

Solution:

Question 5.

Find the area of the parallelogram whose two adjacent sides are determined by the vectors \(\hat{i}+2 \hat{j}+3 \hat{k}\) and \(3 \hat{i}-2 \hat{j}+\hat{k}\)

Solution:

Question 6.

Find the area of the triangle whose vertices are A(3, -1, 2), B(1, -1, -3) and C(4, -3, 1)

Solution:

A = (3, -1, 2); B = (1, -1, -3) and C = (4, -3, 1)

Question 7.

If \(\vec{a}, \vec{b}, \vec{c}\) are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \(\frac{1}{2}|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|\). Also deduce the condition for collinearity of the points A, B, C

Solution:

If the points A, B, C are collinear, then the area of ∆ABC = 0.

Question 8.

For any vector \(\vec{a}\) prove that

Solution:

Question 9.

Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors such that \(\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{b}}=\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{c}}=\mathbf{0}\) and the angle between \(\vec{b} \text { and } \vec{c} \text { is } \frac{\pi}{3}\). Prove that \(\vec{a}=\pm \frac{2}{\sqrt{3}}(\vec{b} \times \vec{c})\)

Solution:

Given \(|\vec{a}|=|\vec{b}|=|\vec{c}|\) = 1

Question 10.

Find the angle between the vectors using vector product

Solution:

The angle between \(\vec{a}\) and \(\vec{b}\) using vector product is given by

### Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.4 Additional Problems

Question 1.

Solution:

Question 2.

If \(\vec{a}\), \(\vec{b}\) are any two vectors, then prove that

Solution:

Question 3.

Find the angle between the vectors by using cross product.

Solution:

Question 4.

Find the vector of magnitude 6 which are perpendicular to both the vectors

Solution:

Question 5.

Find the vectors whose length 5 which are perpendicular to the vectors

Solution:

Question 6.

Solution:

Given \(\vec{a} \times \vec{b}=\vec{c} \times \vec{d}\) and \(\vec{a} \times \vec{c}=\vec{b} \times \vec{d}\)

Question 7.

Find the angle between two vectors \(\vec{a}\) and \(\vec{b}\) if \(|\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{b}}|=\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{b}}\)

Solution:

Question 8.

Solution:

Question 9.

If find the angle between \(\vec{a}\) and \(\vec{b}\)

Solution: