Let $\overrightarrow{\mathrm{a}}=\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}},\overrightarrow{\mathrm{b}}=\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be a vector such that $\left(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right)\times \overrightarrow{\mathrm{c}}=\overrightarrow{0} \mathrm{and} \overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=5$ . Then, the value of $3\left(\overrightarrow{c}.\overrightarrow{\mathrm{a}}\right)$ is equal to____.

We are given the following vectors:

• a = i - 2j + 3k
• b = i + j + k

We are also given that:

• a + b × c = 0
• b ⋅ c = 5

Our goal is to find the value of 3c ⋅ a.

Solution:

Step 1: Given the equation a + b × c = 0, we can express c as:

b × c = -a    

Let c = λa + b, where λ is a scalar constant. Substituting this into the equation:

b × (λa + b) = -a    

Step 2: Now, simplify the cross product. Since b × b = 0 (the cross product of any vector with itself is zero), we have:

λ(b × a) = -a    

Step 3: Now we calculate the cross product b × a. Using the determinant formula for the cross product:

b × a = |i  j  k|                                           |1  1  1|                                           |1 -2  3|    

Expanding the determinant:

b × a = (1 × 3 - 1 × (-2))i - (1 × 3 - 1 × 1)j + (1 × (-2) - 1 × 1)k    

Which simplifies to:

b × a = (3 + 2)i - (3 - 1)j + (-2 - 1)k        = 5i - 2j - 3k    

Step 4: Substituting this back into the equation:

λ(5i - 2j - 3k) = -a    

Step 5: Now substitute the value of a:

λ(5i - 2j - 3k) = -(i - 2j + 3k)    

Equating the coefficients of i, j, and k on both sides, we get:

• For the i component: λ × 5 = -1
• For the j component: λ × (-2) = 2
• For the k component: λ × (-3) = -3

Solving for λ from the j component equation:

λ = 1    

Step 6: Now substitute λ = 1 into the expression for c:

c = λ(a) + b = 1(a) + b        = i - 2j + 3k + i + j + k        = 2i - j + 4k    

Step 7: Finally, we calculate 3c ⋅ a. First, calculate the dot product:

c ⋅ a = (2i - j + 4k) ⋅ (i - 2j + 3k)        = 2(1) + (-1)(-2) + 4(3)        = 2 + 2 + 12        = 16    

Multiplying by 3:

3(c ⋅ a) = 3 × 16 = 48    

Final Answer:

The value of 3c ⋅ a is 48.