Question

Let $ \bar{a} = \bar{i} + \bar{j} + \bar{k} $, $ \bar{b} = \bar{i} + \bar{j} - 2\bar{k} $, $ \bar{c} = \bar{i} - 2\bar{j} + 3\bar{k} $ and $ \bar{d} = -4\bar{i} + 5\bar{j} - 3\bar{k} $ be four vectors. If $ \bar{d} = x(\bar{b} \times \bar{c}) - \frac{7}{9}(\bar{c} \times \bar{a}) + z(\bar{a} \times \bar{b}) $, then $ x = $

• A. $ -\frac{7}{9} $
• B. $ \frac{2}{9} $
• C. $ \frac{23}{9} $
• D. 2

Hint:

Calculate the cross products carefully. Equate the coefficients of the unit vectors to form linear equations and solve for the required scalar.

Answer 1

The Correct Option is B

Step 1: Calculate the cross products.
$ \bar{b} \times \bar{c} = -\bar{i} - 5\bar{j} - 3\bar{k} $ $ \bar{c} \times \bar{a} = -5\bar{i} + 2\bar{j} + 3\bar{k} $ $ \bar{a} \times \bar{b} = -3\bar{i} + 3\bar{j} $
Step 2: Substitute into the equation for $ \bar{d $.}
$ -4\bar{i} + 5\bar{j} - 3\bar{k} = x(-\bar{i} - 5\bar{j} - 3\bar{k}) - \frac{7}{9}(-5\bar{i} + 2\bar{j} + 3\bar{k}) + z(-3\bar{i} + 3\bar{j}) $
Step 3: Equate coefficients of $ \bar{k $.}
$ -3 = -3x - \frac{7}{3} $
Step 4: Solve for $ x $.
$ -3 + \frac{7}{3} = -3x $ $ -\frac{2}{3} = -3x $ $ x = \frac{2}{9} $
Step 5: Conclusion.
The value of $ x $ is $ \frac{2}{9} $.