Question

find the scalar and vector product of two vectors A vector is equal to 3 i- 4j + 5 k and B vector -2 i + j - 3 k

Answer 1

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Answer 2

Answer:

The scalar and vector product of two vectors are $-25\mbox{units}$ and $7\hat i-\hat j-5\hat k$ respectively.

Explanation:

Given two vectors:

$\vec A=3\hat i- 4\hat j + 5\hat k$ and $\vec B=-2\hat i+ \hat j -3\hat k$

Multiplication of vectors is done in two ways: scalar product and vector product.

Scalar Product:

• The scalar product of two vectors is given by the sum of the product of the corresponding components of the vectors.
• If angle between the vectors is given, then the scalar product is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them.
• $\vec A . \vec B= |\vec A|| \vec B|cos\theta$
• The result of the scalar product is a scalar quantity.

Applying to the given vectors, the scalar product is

$\vec A . \vec B= (3\hat i- 4\hat j + 5\hat k)(-2\hat i+ \hat j -3\hat k)$

       $= 3\times(-2)- 4\times1+ 5\times-(3)$

       $= -6- 4-1 5=-25\mbox{units}$

Vector product:

• The magnitude of the vector product of two vectors is given by the area of the parallelogram between them and the direction is given by the right-hand thumb rule.
• If angle between the vectors is given, then the vector product is equal to the product of the magnitudes of the two vectors and the sine of the angle between them.

        $\vec A\times\vec B= |\vec A|| \vec B|sin\theta$

• The result of the vector product is a vector quantity.

Applying the vector product on the given vectors,

$\vec A\times\vec B=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\3&- 4&5\\-2&1&-3\end{array}\right|$

$=((-4)\times(-3)-5\times1)\hat i-(3\times(-3)-5\times(-2))\hat j+(3\times1-(-4)\times(-2))\hat k$

$=(12-5)\hat i-(-9+10)\hat j+(3-8)\hat k$

$=7\hat i-\hat j-5\hat k$

Therefore, the scalar and vector product of two vectors are $-25\mbox{units}$ and $7\hat i-\hat j-5\hat k$ respectively.

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