Question

Consider the following statements in respect of a vector \(\vec c=\vec a+\vec b\), where \(|\vec a|=|\vec b|\ne0\):

1. \(\vec c\) is perpendicular to \((\vec a-\vec b).\)

2. \(\vec c\) is perpendicular to \(\vec a \times \vec b.\)

Which of the above statement is/are correct?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 3 : Both 1 and 2

CONCEPT:

• The scalar triple product of three vectors is zero if any two of them are equal
• If \(\vec a \) is perpendicular to the vectors \(\vec b\) then \(\vec a \cdot \vec b = 0\)

CALCULATION:

Given:  \(\vec c=\vec a+\vec b\), where \(|\vec a|=|\vec b|\ne0\)

Statement 1: \(\vec c\) is perpendicular to \((\vec a-\vec b).\)

First let's find out \(\vec c \cdot (\vec a-\vec b) = (\vec a + \vec b) \cdot (\vec a-\vec b)\)

⇒ \((\vec a + \vec b) \cdot (\vec a-\vec b) = |\vec a|^2 - \vec a \cdot \vec b + \vec b \cdot \vec a - |\vec b|^2\)

As we know that, \(\vec a \cdot \vec b = \vec b \cdot \vec a\)

⇒ \((\vec a + \vec b) \cdot (\vec a-\vec b) = |\vec a|^2 - |\vec b|^2\)

∵ It is given that \(|\vec a|=|\vec b|\ne0\)

⇒ \((\vec a + \vec b) \cdot (\vec a-\vec b) = |\vec a|^2 - |\vec b|^2 = 0\)

⇒ \(\vec c \cdot (\vec a-\vec b) = 0\)

Hence, statement 1 is true.

Statement 2: \(\vec c\) is perpendicular to \(\vec a \times \vec b.\)

First let's find out \(\vec c \cdot (\vec a \times \vec b) = (\vec a + \vec b) \cdot (\vec a \times \vec b)\)

⇒ \((\vec a + \vec b) \cdot (\vec a \times \vec b) = \vec a \cdot (\vec a \times \vec b) + \vec b \cdot (\vec a \times \vec b)\)

As we know that, the scalar triple product of three vectors is zero if any two of them are equal

⇒ \((\vec a + \vec b) \cdot (\vec a \times \vec b) = 0\)

Hence, statement 2 is true.

Hence, the correct option is 3.