find a vector multiplied by b vector and a vector cross b vector if a vector = i hat+2j hat+ k hat and magnitude of b = 3 acting along c vector = i hat+ j hat+ k hat
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In the definition of the dot product, the direction of angle [latex] \phi [/latex] does not matter, and [latex] \phi [/latex] can be measured from either of the two vectors to the other because [latex] \text{cos}\,\phi =\text{cos}\,(\text{−}\phi )=\text{cos}\,(2\pi -\phi ) [/latex]. The dot product is a negative number when [latex] 90\text{°}<\phi \le 180\text{°} [/latex] and is a positive number when [latex] 0\text{°}\le \phi <90\text{°} [/latex]. Moreover, the dot product of two parallel vectors is [latex] \overset{\to }{A}·\overset{\to }{B}=AB\,\text{cos}\,0\text{°}=AB [/latex], and the dot product of two antiparallel vectors is [latex] \overset{\to }{A}·\overset{\to }{B}=AB\,\text{cos}\,180\text{°}=\text{−}AB [/latex]. The scalar product of two orthogonal vectors vanishes: [latex] \overset{\to }{A}·\overset{\to }{B}=AB\,\text{cos}\,90\text{°}=0 [/latex]. The scalar product of a vector with itself is the square of its magnitude:
[latex] {\overset{\to }{A}}^{2}\equiv \overset{\to }{A}·\overset{\to }{A}=AA\,\text{cos}\,0\text{°}={A}^{2}. [/latex]
Figure a: vectors A and B are shown tail to tail. A is longer than B. The angle between them is phi. Figure b: Vector B is extended using a dashed line and another dashed line is drawn from the head of A to the extension of B, perpendicular to B. A sub perpendicular is equal to A magnitude times cosine phi and is the distance from the vertex where the tails of A and B meet to the location where the perpendicular from A to B meets the extension of B. Figure c: A dashed line is drawn from the head of B to A, perpendicular to A. The distance from the tails of A and B to where the dashed line meets B is B sub perpendicular and is equal to magnitude B times cosine phi.
Figure 2.27 The scalar product of two vectors. (a) The angle between the two vectors. (b) The orthogonal projection [latex] {A}_{\perp } [/latex] of vector [latex] \overset{\to }{A} [/latex] onto the direction of vector [latex] \overset{\to }{B} [/latex]. (c) The orthogonal projection [latex] {B}_{\perp } [/latex] of vector [latex] \overset{\to }{B} [/latex] onto the direction of vector [latex] \overset{\to }{A} [/latex].
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Answer:
In the definition of the dot product, the direction of angle [latex] \phi [/latex] does not matter, and [latex] \phi [/latex] can be measured from either of the two vectors to the other because [latex