Given vectors a = 2i + 5j − k and b = 3i − 2j + 4k, find a x b, b x a, and (a + b) x (a − b).

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Given the vectors: a = 2i + 5j − k; b = 3i − 2j + 4k We need to find (i) a × b, (ii) b × a, and (iii) (a + b) × (a − b). i) Find a × b. Step 1: Set up the determinant for the cross product. The cross product a × b is calculated as: a × b = i & j & k 2 & 5 & −1 3 & −2 & 4 Step 2: Calculate the components of the cross product. a × b = ( (5)(4) − (−1)(−2) )i − ( (2)(4) − (−1)(3) )j + ( (2)(−2) − (5)(3) )k; a × b = (20 − 2)i − (8 − (−3))j + (−4 − 15)k; a × b = 18i − (8 + 3)j − 19k; a × b = 18i − 11j − 19k The result for (i) is 18i − 11j − 19k. ii) Find b × a. Step 1: Use the property of cross products that b × a = −(a × b). Using the result from part (i): b × a = −(18i − 11j − 19k); b × a = −18i + 11j + 19k The result for (ii) is −18i + 11j + 19k. iii) Find (a + b) × (a − b). Step 1: Calculate a + b. a + b = (2i + 5j − k) + (3i − 2j + 4k); a + b = (2+3)i + (5−2)j + (−1+4)k; a + b = 5i + 3j + 3k Step 2: Calculate a − b. a − b = (2i + 5j − k) − (3i − 2j + 4k); a − b = (2−3)i + (5−(−2))j + (−1−4)k; a − b = −1i + 7j − 5k Step 3: Set up the determinant for the cross product of (a + b) and (a − b). (a + b) × (a − b) = i & j & k 5 & 3 & 3 −1 & 7 & −5 Step 4: Calculate the components of the cross product. (a + b) × (a − b) = ( (3)(−5) − (3)(7) )i − ( (5)(−5) − (3)(−1) )j + ( (5)(7) − (3)(−1) )k; (a + b) × (a − b) = (−15 − 21)i − (−25 − (−3))j + (35 − (−3))k; (a + b) × (a − b) = −36i − (−25 + 3)j + (35 + 3)k; (a + b) × (a − b) = −36i − (−22)j + 38k; (a + b) × (a − b) = −36i + 22j + 38k The result for (iii) is −36i + 22j + 38k.

Given vectors a = 2i + 5j − k and b = 3i − 2j + 4k, find a x b, b x a, and (a + b) x (a - b).

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Step-by-step solution

Part i· 2 steps

Set up the determinant for the cross product.

Calculate the components of the cross product.

Part ii· 1 step
$-18\mathbf{i} + 11\mathbf{j} + 19\mathbf{k}$

Use the property of cross products that $\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$ .

Part iii· 4 steps
$-36\mathbf{i} + 22\mathbf{j} + 38\mathbf{k}$

Calculate $\mathbf{a} + \mathbf{b}$ .

Calculate $\mathbf{a} - \mathbf{b}$ .

Set up the determinant for the cross product of $(\mathbf{a} + \mathbf{b})$ and $(\mathbf{a} - \mathbf{b})$ .

Calculate the components of the cross product.

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Given vectors a = 2i + 5j − k and b = 3i − 2j + 4k, find a x b, b x a, and (a + b) x (a - b).

Answers

Step-by-step solution

Part i· 2 steps

Set up the determinant for the cross product.

Calculate the components of the cross product.

Part ii· 1 step−18i+11j+19k-18\mathbf{i} + 11\mathbf{j} + 19\mathbf{k}−18i+11j+19k

Use the property of cross products that b×a=−(a×b)\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})b×a=−(a×b).

Part iii· 4 steps−36i+22j+38k-36\mathbf{i} + 22\mathbf{j} + 38\mathbf{k}−36i+22j+38k

Calculate a+b\mathbf{a} + \mathbf{b}a+b.

Calculate a−b\mathbf{a} - \mathbf{b}a−b.

Set up the determinant for the cross product of (a+b)(\mathbf{a} + \mathbf{b})(a+b) and (a−b)(\mathbf{a} - \mathbf{b})(a−b).

Calculate the components of the cross product.

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Part ii· 1 step
$-18\mathbf{i} + 11\mathbf{j} + 19\mathbf{k}$

Use the property of cross products that $\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$ .

Part iii· 4 steps
$-36\mathbf{i} + 22\mathbf{j} + 38\mathbf{k}$

Calculate $\mathbf{a} + \mathbf{b}$ .

Calculate $\mathbf{a} - \mathbf{b}$ .

Set up the determinant for the cross product of $(\mathbf{a} + \mathbf{b})$ and $(\mathbf{a} - \mathbf{b})$ .