Question 44 of 68intermediate🔧 ApplyLong Answer4 marks

Let the vectors a⃗, b⃗, c⃗ be given as a₁î + a₂ĵ + a₃k̂, b₁î + b₂ĵ + b₃k̂, c₁î + c₂ĵ + c₃k̂. Then show that a⃗ × (b⃗ + c⃗) = a⃗ × b⃗ + a⃗ × c⃗.

Correct Answer

Given vectors:

a⃗=a1i^+a2j^+a3k^\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} b⃗=b1i^+b2j^+b3k^\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} c⃗=c1i^+c2j^+c3k^\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}

LHS: a⃗×(b⃗+c⃗)\vec{a} \times (\vec{b} + \vec{c})

First, using the sum property:

b⃗+c⃗=(b1+c1)i^+(b2+c2)j^+(b3+c3)k^\vec{b} + \vec{c} = (b_1 + c_1)\hat{i} + (b_2 + c_2)\hat{j} + (b_3 + c_3)\hat{k}

Now applying cross product in determinant form:

a⃗×(b⃗+c⃗)=∣i^j^k^a1a2a3b1+c1b2+c2b3+c3∣\vec{a} \times (\vec{b} + \vec{c}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 + c_1 & b_2 + c_2 & b_3 + c_3 \end{vmatrix}

Using the property that determinant of sum equals sum of determinants:

=∣i^j^k^a1a2a3b1b2b3∣+∣i^j^k^a1a2a3c1c2c3∣= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} + \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

RHS: a⃗×b⃗+a⃗×c⃗\vec{a} \times \vec{b} + \vec{a} \times \vec{c}

=∣i^j^k^a1a2a3b1b2b3∣+∣i^j^k^a1a2a3c1c2c3∣= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} + \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Since LHS = RHS, we have proved:

a⃗×(b⃗+c⃗)=a⃗×b⃗+a⃗×c⃗\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
Exercise: EXERCISE 10.4 | Q: 7 | (Page 31)
For More Understanding

Explanation

This question tests the distributive property of vector cross product over vector addition. The solution uses the component form of vectors and the determinant representation of cross product provided in the textbook context. The key step is applying the determinant property that allows splitting a row sum into separate determinants, which directly yields the required result.

Solution Steps

  1. Step 1: Write vectors a, b, c in component form

  2. Step 2: Find b + c using vector addition property

  3. Step 3: Express a × (b + c) as a determinant

  4. Step 4: Split the determinant using row addition property

  5. Step 5: Recognize each determinant as a × b and a × c respectively

  6. Step 6: Conclude LHS = RHS

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