Question

If vector a = i + j + 2k and vector b = 3i + 2j - k, then the value of vector(a + 3b).(2a - b) = (a) 15 (b) -15 (c) 18 (d) -18

Answer 1

Answer:

If vector a = i + j + 2k and vector b = 3i + 2j - k, then the value of vector(a + 3b).(2a - b) = 35i - 49j + 7k

Step-by-step explanation:

To multiply the vectors, let's first get the values of the brackets.

3 b = 3(3i + 2j - k) = 9i + 6j - 3k

2a = 2(i + j + 2k) = 2i + 2j + 4k

Solving the brackets we have:

a + 3b = (i + j + 2k) + (9i + 6j - 3k) = 10i + 7j - k

2a - b = (2i + 2j + 4k) - (3i + 2j - k) = -i + 0j + 5k

Now we have:

(10i + 7j - k) (-i + 0j + 5k)

To multiply this, we need to get the dot product of the vectors.

To get this, we write it in matrix form.

So we have:

$\left[\begin{array}{ccc}i&j&k\\10&7&-1\\-1&0&5\end{array}\right]$

The cross product of this vector is given by the determinant of the matrix

This is given by:

= i(7 × 5 - 0 × -1) -j(10 × 5 - (-1 × -1)) + k(10 × 0 - (-1 × 7)

= i(35) -j(49) + k(7)

= 35i - 49j + 7k

Answer 2

Answer:

If vector a = i + j + 2k and vector b = 3i + 2j - k, then the value of vector(a + 3b).(2a - b) = 35i - 49j + 7k

Step-by-step explanation:

To multiply the vectors, let's first get the values of the brackets.

3 b = 3(3i + 2j - k) = 9i + 6j - 3k

2a = 2(i + j + 2k) = 2i + 2j + 4k

Solving the brackets we have:

a + 3b = (i + j + 2k) + (9i + 6j - 3k) = 10i + 7j - k

2a - b = (2i + 2j + 4k) - (3i + 2j - k) = -i + 0j + 5k

Now we have:

(10i + 7j - k) (-i + 0j + 5k)

To multiply this, we need to get the dot product of the vectors.

To get this, we write it in matrix form.

So we have:

The cross product of this vector is given by the determinant of the matrix

This is given by:

= i(7 × 5 - 0 × -1) -j(10 × 5 - (-1 × -1)) + k(10 × 0 - (-1 × 7)

= i(35) -j(49) + k(7)

= 35i - 49j + 7k

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