Question

which is the true of the following ?
(A)Any vector \(\overrightarrow{r}\) in space can be written as \((\overrightarrow{r}.\hat{i})\hat{i}+(\overrightarrow{r}.\hat{j})\hat{j}+(\overrightarrow{r}.\hat{k})\hat{k}\) 
(B)If  \(\overrightarrow{a}\)  is perpendicular to  \(\overrightarrow{b}\)\(|\overrightarrow{a}+\overrightarrow{b}|^2=|\overrightarrow{a}|^2+|\overrightarrow{b}|^2\) 
 
(C)If \(|\overrightarrow{a}|=2,|\overrightarrow{b}|=1 \) and \(\overrightarrow{a}.\overrightarrow{b}=1 \) ,the value of \((3\overrightarrow{a}-5\overrightarrow{b}).(2\overrightarrow{a}+7\overrightarrow{b})\) ia 1
(D) \(\overrightarrow{a}=5\hat{i}-\hat{j}-3\hat{k}\) and \(\overrightarrow{b}=\hat{i}+3\hat{j}-5\hat{k}\), is the angle between \(\overrightarrow{a}+\overrightarrow{b}\) and \(\overrightarrow{a}-\overrightarrow{b}\) is \(60\degree\)
Choose the correct answer from the options given below:
 

• (A) and (B) Only
• (B) Only
• (C) Only
• (D) Only

Answer 1

The Correct Option is A

To solve this problem, we'll analyze each given option:
(A) Any vector \(\overrightarrow{r}\) in space can be expressed as \((\overrightarrow{r} \cdot \hat{i})\hat{i} + (\overrightarrow{r} \cdot \hat{j})\hat{j} + (\overrightarrow{r} \cdot \hat{k})\hat{k}\). This is true because any vector can be decomposed into its components along the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). Thus, option (A) is correct.
(B) If \(\overrightarrow{a}\) is perpendicular to \(\overrightarrow{b}\), then \(|\overrightarrow{a} + \overrightarrow{b}|^2 = |\overrightarrow{a}|^2 + |\overrightarrow{b}|^2\). This follows directly from the Pythagorean theorem, as the dot product \(\overrightarrow{a} \cdot \overrightarrow{b} = 0\). Therefore, option (B) is correct.
(C) For \(|\overrightarrow{a}| = 2\), \(|\overrightarrow{b}| = 1\), and \(\overrightarrow{a} \cdot \overrightarrow{b} = 1\), the expression \((3\overrightarrow{a} - 5\overrightarrow{b}) \cdot (2\overrightarrow{a} + 7\overrightarrow{b})\) simplifies to \(2(3|\overrightarrow{a}|^2 + 7(\overrightarrow{a} \cdot \overrightarrow{b}) - 5(\overrightarrow{b} \cdot \overrightarrow{a}) - 35|\overrightarrow{b}|^2)\). Substituting the given values results in a different value from 1; thus, (C) is incorrect.
(D) Given \(\overrightarrow{a} = 5\hat{i} - \hat{j} - 3\hat{k}\) and \(\overrightarrow{b} = \hat{i} + 3\hat{j} - 5\hat{k}\), the angle between \(\overrightarrow{a} + \overrightarrow{b}\) and \(\overrightarrow{a} - \overrightarrow{b}\) is not \(60^\degree\). This can be verified by computing the individual vectors and their dot product, checking against the cosine rule for angles. Thus, (D) is incorrect.
The correct answer, therefore, is (A) and (B) Only.