Question

5. Given that vectors
a→=5i^−2j^+3k^
b→=3i^+j^−2k^
c→=i^−3j^+4k^
, calculate the scalar triple product
a⃗ ⋅(b⃗ ×c⃗ )

6. Given that vectors
a→=5i^−2j^+3k^,
b→=3i^+j^−2k^
and
c→=i^−3j^+4k^
, calculate the vector triple product
a⃗ ×(b⃗ ×c⃗ )

Answer 1

Answer on Question #61859 – Math – Vector Calculus

Question

5. Given that vectors

$$a \rightarrow = 5i - 2j + 3k$$ $$b \rightarrow = 3i + j - 2k$$ $$c \rightarrow = i - 3j + 4k,$$

calculate the scalar triple product

$$\vec{a} \cdot (\vec{b} \times \vec{c})$$

Solution

The scalar triple product is

$$\begin{aligned} \vec{a} \cdot (\vec{b} \times \vec{c}) &= \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} = \det \begin{bmatrix} 5 & -2 & 3 \\ 3 & 1 & -2 \\ 1 & -3 & 4 \end{bmatrix} \\ &= 5 \cdot 1 \cdot 4 + (-2) \cdot (-2) \cdot 1 + 3 \cdot 3 \cdot (-3) - 1 \cdot 1 \cdot 3 - (-3) \cdot (-2) \cdot 5 - 4 \cdot 3 \cdot (-2) = \\ &= 20 + 4 - 27 - 3 - 30 + 24 = -12 \end{aligned}$$

Answer: -12

Question

6. Given that vectors

$$a \rightarrow = 5i - 2j + 3k,$$ $$b \rightarrow = 3i + j - 2k$$

and

$$c \rightarrow = i - 3j + 4k,$$

calculate the vector triple product

$$\vec{a} \times (\vec{b} \times \vec{c})$$

Solution

$$\begin{aligned} (\vec{b} \times \vec{c}) &= \det \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & -2 \\ 1 & -3 & 4 \end{bmatrix} = \hat{i} \left| \begin{array}{cc} 1 & -2 \\ -3 & 4 \end{array} \right| - \hat{j} \left| \begin{array}{cc} 3 & -2 \\ 1 & 4 \end{array} \right| + \hat{k} \left| \begin{array}{cc} 3 & 1 \\ 1 & -3 \end{array} \right| = \hat{i} (1 \cdot 4 - (-3) \cdot (-2)) - \\ j (3 \cdot 4 - 1 \cdot (-2)) + \hat{k} (3 \cdot (-3) - 1 \cdot 1) &= -2\hat{i} - 14\hat{j} - 10\hat{k}. \end{aligned}$$

The vector triple product is

$$\begin{array}{l} \hat{a} \times (\vec{b} \times \vec{c}) = \det \left[ \begin{array}{ccc} i & j & k \\ 5 & -2 & 3 \\ -2 & -14 & -10 \end{array} \right] = \hat{i} \left| \begin{array}{cc} -2 & 3 \\ -14 & -10 \end{array} \right| - \hat{j} \left| \begin{array}{cc} 5 & 3 \\ -2 & -10 \end{array} \right| + \hat{k} \left| \begin{array}{cc} 5 & -2 \\ -2 & -14 \end{array} \right| = \\ = \hat{i} \left( (-2) \cdot (-10) - (-14) \cdot 3 \right) - j (5 \cdot (-10) - (-2) \cdot 3) + \hat{k} \left(5 \cdot (-14) - (-2) \cdot (-2) \right) \\ = 62 \hat{i} + 44 \hat{j} - 74 \hat{k} \end{array}$$

Answer: $62\hat{i} + 44\hat{j} - 74\hat{k}$.

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