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11. If [tex]\vec{A}=4 \vec{i}+6 \vec{j}+3 \vec{k}[/tex] and [tex]\vec{B}=3 \vec{i}+4 \vec{j}+5 \vec{k}[/tex], then [tex]\vec{A} \cdot \vec{B}=[/tex]

1) 51

2) 41

3) 34


Sagot :

To find the dot product of two vectors [tex]\(\vec{A} = 4\vec{i} + 6\vec{j} + 3\vec{k}\)[/tex] and [tex]\(\vec{B} = 3\vec{i} + 4\vec{j} + 5\vec{k}\)[/tex], we use the formula for the dot product of two vectors:

[tex]\[ \vec{A} \cdot \vec{B} = A_i B_i + A_j B_j + A_k B_k \][/tex]

Where:
- [tex]\(A_i\)[/tex], [tex]\(A_j\)[/tex], and [tex]\(A_k\)[/tex] are the components of vector [tex]\(\vec{A}\)[/tex].
- [tex]\(B_i\)[/tex], [tex]\(B_j\)[/tex], and [tex]\(B_k\)[/tex] are the components of vector [tex]\(\vec{B}\)[/tex].

In this problem, we have:
[tex]\[ A_i = 4, \quad A_j = 6, \quad A_k = 3 \][/tex]
[tex]\[ B_i = 3, \quad B_j = 4, \quad B_k = 5 \][/tex]

Now, substitute these values into the formula:

[tex]\[ \vec{A} \cdot \vec{B} = (4 \cdot 3) + (6 \cdot 4) + (3 \cdot 5) \][/tex]

Calculate each term individually:
[tex]\[ 4 \cdot 3 = 12 \][/tex]
[tex]\[ 6 \cdot 4 = 24 \][/tex]
[tex]\[ 3 \cdot 5 = 15 \][/tex]

Now add these results together:
[tex]\[ \vec{A} \cdot \vec{B} = 12 + 24 + 15 = 51 \][/tex]

Therefore, the dot product [tex]\(\vec{A} \cdot \vec{B}\)[/tex] is 51. The correct answer is:

1) 51
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