How to Find Scalar Multiplication of Vectors?
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To multiply a vector by a scalar, multiply each component by a scalar. The product of a scalar with a vector is always a vector.
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A step-by-step guide to scalar multiplication of vectors
The scalar component of the vector is multiplied by the scalar component of each component of the vector.
If \(u⃗=(u_1, u_2)\) has a magnitude of \(|u⃗ |\) and a direction of \(d\), then:
\(\color{blue}{|nu⃗ |=n(u_1,u_2)=(nu_1,nu_2)}\)
where \(n\) is a positive real integer, the magnitude is \(|nu⃗ |\) and the direction is \(d\).
If \(n\) is negative, the direction of \(n\)u is the total opposite of \(d\).
Properties of Scalar Multiplication:
Allow u and v to be vectors, and \(c\) and \(d\) to be scalars. Then the properties listed below are true:
Scalar Multiplication of Vectors – Example 1:
If u\(=(-2, 3)\), find \(6\)u.
\(6\)u\(=6(-2,3)\)
\(=6(-2),6(3)\)
\(=(-12,18)\)
Scalar Multiplication of Vectors – Example 2:
If a\(=3i+j+2k\), find \(5\)a.
\(5\)a\(=5(3i+j+2k)\)
\(=5(3i)+5(j)+5(2k)\)
\(=15i+5j+10k\)
Exercises for Scalar Multiplication of Vectors
- If a\(=(-1,-8)\), find \(4\)a.
- If v\(=6i-5j+4k\), find \(3\)v.
- If a\(=7, -3\), find \(-5\)a.
- If v\(=i-7j-5k\), find \(-6\)v.
- \(\color{blue}{4a=(-4,-32)}\)
- \(\color{blue}{3v=18i-15j+12k}\)
- \(\color{blue}{-5a=(-35,15)}\)
- \(\color{blue}{-6v=-6i+42j+35k}\)
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